Metamath Proof Explorer - mmtheorems38

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Statement List for Metamath Proof Explorer - 3701-3800 - Page 38 of 58

Type	Label	Description
Statement
Theorem	cfval 3701	Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number A is the cardinality (size) of the smallest unbounded subset y of the ordinal number. Unbounded means that for every member of A, there is a member of y that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is.
⊢ (A ∈ On → (cf ‘A) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ ∀z ∈ A ∃w ∈ y z ⊆ w))})
Theorem	cffnon 3702	Cofinality is a function on the class of ordinal numbers.
⊢ cf Fn On
Theorem	cfub 3703	An upper bound on cofinality.
⊢ (cf ‘A) ⊆ ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A ⊆ ∪y))}
Theorem	cflim 3704	Value of the cofinality function at a limit ordinal. Part of Definition of cofinality of [Enderton] p. 257.
⊢ ((A ∈ B ∧ Lim A) → (cf ‘A) = ∩{x∣∃y(x = (card ‘y) ∧ (y ⊆ A ∧ A = ∪y))})
Theorem	cf0 3705	Value of the cofinality function at 0. Exercise 2 of [TakeutiZaring] p. 102.
⊢ (cf ‘∅) = ∅
Theorem	cardcf 3706	Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103.
⊢ (card ‘(cf ‘A)) = (cf ‘A)
Theorem	cflecard 3707	Cofinality is bounded by the cardinality of its argument.
⊢ (cf ‘A) ⊆ (card ‘A)
Theorem	cfle 3708	Cofinality is bounded by its argument. Exercise 1 of [TakeutiZaring] p. 102.
⊢ (cf ‘A) ⊆ A
Theorem	cfsuc 3709	Value of the cofinality function at a successor ordinal. Exercise 3 of [TakeutiZaring] p. 102.
⊢ (A ∈ On → (cf ‘suc A) = 1o)
Theorem	cfom 3710	Value of the cofinality function at omega (the set of natural numbers). Exercise 4 of [TakeutiZaring] p. 102.
⊢ (cf ‘ω) = ω
Syntax	ccdn 3711	Extend class definition to include the class of all cardinal numbers.
class Card
Definition	df-cardn 3712	Define the class of all cardinal numbers. The notation "Card" is used in Exercise 5(G) of [JustWeese] p. 174. It should not be confused with the lower-case "card" for the cardinal number function df-card 3623.
⊢ Card = (ω ∪ ran ℵ)
Theorem	elcard 3713	Membership in the class of cardinal numbers.
⊢ (A ∈ Card ↔ (card ‘A) = A)
Syntax	ccda 3714	Extend class definition to include cardinal number addition.
class +c
Definition	df-cda 3715	Define cardinal number addition. Definition of cardinal sum in [Mendelson] p. 258. See cdaval 3717 for its value and a description.
⊢ +c = {⟨⟨x, y⟩, z⟩∣z = ((x × {∅}) ∪ (y × {1o}))}
Theorem	cdavalt 3716	Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258.
⊢ ((A ∈ C ∧ B ∈ D) → (A +c B) = ((A × {∅}) ∪ (B × {1o})))
Theorem	cdaval 3717	Value of cardinal addition. Definition of cardinal sum in [Mendelson] p. 258. For cardinal arithmetic, we follow Mendelson. Rather than defining operations restricted to cardinal numbers, we use this disjoint union operation for addition, while cross product and set exponentiation stand in for cardinal multiplication and exponentiation. Equinumerosity and dominance serve the roles of equality and ordering. If we wanted to, we could easily convert our theorems to actual cardinal number operations via carden 3638, carddom 3642, and cardsdom 3643. The advantage of Mendelson's approach is that we can directly use many equinumerosity theorems that we already have available.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A +c B) = ((A × {∅}) ∪ (B × {1o}))
Theorem	uncdadom 3718	Cardinal addition dominates union.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A ∪ B) ≼ (A +c B)
Theorem	cdaen 3719	Cardinal addition of equinumerous sets. Exercise 4.56(b) of [Mendelson] p. 258.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    &   ⊢ D ∈ V    ⇒   ⊢ ((A ≈ B ∧ C ≈ D) → (A +c C) ≈ (B +c D))
Theorem	cda0en 3720	Cardinal addition with cardinal zero (the empty set). Part (a1) of proof of Theorem 6J of [Enderton] p. 143.
⊢ A ∈ V    ⇒   ⊢ (A +c ∅) ≈ A
Theorem	cda1en 3721	Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143.
⊢ A ∈ V    ⇒   ⊢ (A +c 1o) ≈ suc (card ‘A)
Theorem	xp1en 3722	One times a cardinal number.
⊢ A ∈ V    ⇒   ⊢ (A × 1o) ≈ A
Theorem	xp2cda 3723	Two times a cardinal number. Exercise 4.56(g) of [Mendelson] p. 258.
⊢ A ∈ V    ⇒   ⊢ (A × 2o) = (A +c A)
Theorem	cdacomen 3724	Commutative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ (A +c B) ≈ (B +c A)
Theorem	cdaassen 3725	Associative law for cardinal addition. Exercise 4.56(c) of [Mendelson] p. 258.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ ((A +c B) +c C) ≈ (A +c (B +c C))
Theorem	xpcdaen 3726	Cardinal multiplication distributes over cardinal addition. Theorem 6I(3) of [Enderton] p. 142.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ (A × (B +c C)) ≈ ((A × B) +c (A × C))
Theorem	cdadom1 3727	Ordering law for cardinal addition. Exercise 4.56(f) of [Mendelson] p. 258.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ (A ≼ B → (A +c C) ≼ (B +c C))
Theorem	cdadom2 3728	Ordering law for cardinal addition. Theorem 6L(a) of [Enderton] p. 149.
⊢ A ∈ V    &   ⊢ B ∈ V    &   ⊢ C ∈ V    ⇒   ⊢ (A ≼ B → (C +c A) ≼ (C +c B))
Theorem	cdadom3 3729	A set is dominated by its cardinal sum with another.
⊢ A ∈ V    &   ⊢ B ∈ V    ⇒   ⊢ A ≼ (A +c B)
Theorem	cdafi 3730	The cardinal sum of two finite sets is finite.
⊢ ((A ≺ ω ∧ B ≺ ω) → (A +c B) ≺ ω)
Theorem	cdainf 3731	A set is infinite iff the cardinal sum with itself is infinite.
⊢ A ∈ V    ⇒   ⊢ (ω ≼ A ↔ ω ≼ (A +c A))
Theorem	nd1 3732	Lemma for proving conditionless ZFC axioms.
⊢ (∀x x = y → ¬ ∀x y ∈ z)
Theorem	nd2 3733	Lemma for proving conditionless ZFC axioms.
⊢ (∀x x = y → ¬ ∀x z ∈ y)
Theorem	nd3 3734	Lemma for proving conditionless ZFC axioms.
⊢ (∀x x = y → ¬ ∀z x ∈ y)
Theorem	nd4 3735	Lemma for proving conditionless ZFC axioms.
⊢ (∀x x = y → ¬ ∀z y ∈ x)
Theorem	nd5 3736	Lemma for proving conditionless ZFC axioms.
⊢ (¬ ∀y y = x → (z = y → ∀x z = y))
Theorem	axextnd 3737	A version of the Axiom of Extensionality with no distinct variable conditions.
⊢ ∃x((x ∈ y ↔ x ∈ z) → y = z)
Theorem	axrepndlem1 3738	Lemma for the Axiom of Replacement with no distinct variable conditions.
⊢ (¬ ∀y y = z → ∃x(∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ y ∧ ∀yφ))))
Theorem	axrepndlem2 3739	Lemma for the Axiom of Replacement with no distinct variable conditions.
⊢ (((¬ ∀x x = y ∧ ¬ ∀x x = z) ∧ ¬ ∀y y = z) → ∃x(∃y∀z(φ → z = y) → ∀z(z ∈ x ↔ ∃x(x ∈ y ∧ ∀yφ))))
Theorem	axrepnd 3740	A version of the Axiom of Replacement with no distinct variable conditions.
⊢ ∃x(∃y∀z(φ → z = y) → ∀z(∀y z ∈ x ↔ ∃x(∀z x ∈ y ∧ ∀yφ)))
Theorem	axunndlem1 3741	Lemma for the Axiom of Union with no distinct variable conditions.
⊢ ∃x∀y(∃x(y ∈ x ∧ x ∈ z) → y ∈ x)
Theorem	axunnd 3742	A version of the Axiom of Union with no distinct variable conditions.
⊢ ∃x∀y(∃x(y ∈ x ∧ x ∈ z) → y ∈ x)
Theorem	axpowndlem1 3743	Lemma for the Axiom of Power Sets with no distinct variable conditions.
⊢ (∀x x = y → (¬ x = y → ∃x∀y(∀x(∃z x ∈ y → ∀y x ∈ z) → y ∈ x)))
Theorem	axpowndlem2 3744	Lemma for the Axiom of Power Sets with no distinct variable conditions.
⊢ (¬ ∀x x = y → (¬ ∀x x = z → (¬ x = y → ∃x∀y(∀x(∃z x ∈ y → ∀y x ∈ z) → y ∈ x))))
Theorem	axpowndlem3 3745	Lemma for the Axiom of Power Sets with no distinct variable conditions.
⊢ (¬ x = y → ∃x∀y(∀x(∃z x ∈ y → ∀y x ∈ z) → y ∈ x))
Theorem	axpowndlem4 3746	Lemma for the Axiom of Power Sets with no distinct variable conditions.
⊢ (¬ ∀y y = x → (¬ ∀y y = z → (¬ x = y → ∃x∀y(∀x(∃z x ∈ y → ∀y x ∈ z) → y ∈ x))))
Theorem	axpownd 3747	A version of the Axiom of Power Sets with no distinct variable conditions.
⊢ (¬ x = y → ∃x∀y(∀x(∃z x ∈ y → ∀y x ∈ z) → y ∈ x))
Theorem	axregndlem1 3748	Lemma for the Axiom of Regularity with no distinct variable conditions.
⊢ (∀x x = z → (x ∈ y → ∃x(x ∈ y ∧ ∀z(z ∈ x → ¬ z ∈ y))))
Theorem	axregndlem2 3749	Lemma for the Axiom of Regularity with no distinct variable conditions.
⊢ (x ∈ y → ∃x(x ∈ y ∧ ∀z(z ∈ x → ¬ z ∈ y)))
Theorem	axregnd 3750	A version of the Axiom of Regularity with no distinct variable conditions.
⊢ (x ∈ y → ∃x(x ∈ y ∧ ∀z(z ∈ x → ¬ z ∈ y)))
Theorem	axinfndlem1 3751	Lemma for the Axiom of Infinity with no distinct variable conditions.
⊢ (∀x y ∈ z → ∃x(y ∈ x ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x))))
Theorem	axinfnd 3752	A version of the Axiom of Infinity with no distinct variable conditions.
⊢ ∃x(y ∈ z → (y ∈ x ∧ ∀y(y ∈ x → ∃z(y ∈ z ∧ z ∈ x))))
Theorem	axacndlem1 3753	Lemma for the Axiom of Choice with no distinct variable conditions.
⊢ (∀x x = y → ∃x∀y∀z(∀x(y ∈ z ∧ z ∈ w) → ∃w∀y(∃w((y ∈ z ∧ z ∈ w) ∧ (y ∈ w ∧ w ∈ x)) ↔ y = w)))
Theorem	axacndlem2 3754	Lemma for the Axiom of Choice with no distinct variable conditions.
⊢ (∀x x = z → ∃x∀y∀z(∀x(y ∈ z ∧ z ∈ w) → ∃w∀y(∃w((y ∈ z ∧ z ∈ w) ∧ (y ∈ w ∧ w ∈ x)) ↔ y = w)))
Theorem	axacndlem3 3755	Lemma for the Axiom of Choice with no distinct variable conditions.
⊢ (∀y y = z → ∃x∀y∀z(∀x(y ∈ z ∧ z ∈ w) → ∃w∀y(∃w((y ∈ z ∧ z ∈ w) ∧ (y ∈ w ∧ w ∈ x)) ↔ y = w)))
Theorem	axacndlem4 3756	Lemma for the Axiom of Choice with no distinct variable conditions.
⊢ ∃x∀y∀z(∀x(y ∈ z ∧ z ∈ w) → ∃w∀y(∃w((y ∈ z ∧ z ∈ w) ∧ (y ∈ w ∧ w ∈ x)) ↔ y = w))
Theorem	axacndlem5 3757	Lemma for the Axiom of Choice with no distinct variable conditions.
⊢ ∃x∀y∀z(∀x(y ∈ z ∧ z ∈ w) → ∃w∀y(∃w((y ∈ z ∧ z ∈ w) ∧ (y ∈ w ∧ w ∈ x)) ↔ y = w))
Theorem	axacnd 3758	A version of the Axiom of Choice with no distinct variable conditions.
⊢ ∃x∀y∀z(∀x(y ∈ z ∧ z ∈ w) → ∃w∀y(∃w((y ∈ z ∧ z ∈ w) ∧ (y ∈ w ∧ w ∈ x)) ↔ y = w))
Theorem	zfcndext 3759	Axiom of Extensionality, reproved from conditionless ZFC version. We use only predicate calculus in the proof.
⊢ (∀z(z ∈ x ↔ z ∈ y) → x = y)
Theorem	zfcndrep 3760	Axiom of Replacement, reproved from conditionless ZFC axioms. We use several results such as visset 1350 that depend on Extensionality, which was already proved in zfcndext 3759.
⊢ (∀w∃y∀z(∀yφ → z = y) → ∃y∀z(z ∈ y ↔ ∃w(w ∈ x ∧ ∀yφ)))
Theorem	zfcndun 3761	Axiom of Union, reproved from conditionless ZFC axioms.
⊢ ∃y∀z(∃w(z ∈ w ∧ w ∈ x) → z ∈ y)
Theorem	zfcndpow 3762	Axiom of Power Sets, reproved from conditionless ZFC axioms. The proof uses the "Axiom of Twoness," dtru 1889.
⊢ ∃y∀z(∀w(w ∈ z → w ∈ x) → z ∈ y)
Theorem	zfcndreg 3763	Axiom of Regularity, reproved from conditionless ZFC axioms..
⊢ (∃y y ∈ x → ∃y(y ∈ x ∧ ∀z(z ∈ y → ¬ z ∈ x)))
Theorem	zfcndinf 3764	Axiom of Infinity, reproved from conditionless ZFC axioms.
⊢ ∃y(x ∈ y ∧ ∀z(z ∈ y → ∃w(z ∈ w ∧ w ∈ y)))
Theorem	zfcndac 3765	Axiom of Choice, reproved from conditionless ZFC axioms.
⊢ ∃y∀z∀w((z ∈ w ∧ w ∈ x) → ∃v∀u(∃t((u ∈ w ∧ w ∈ t) ∧ (u ∈ t ∧ t ∈ y)) ↔ u = v))
Syntax	cnpi 3766	Set of positive integers.
class N
Syntax	cpli 3767	Positive integer addition.
class +N
Syntax	cmi 3768	Positive integer multiplication.
class ·N
Syntax	clti 3769	Positive integer ordering relation.
class <N
Syntax	cplpq 3770	Positive fraction pre-addition.
class +pQ
Syntax	cmpq 3771	Positive fraction pre-multiplication.
class ·pQ
Syntax	ceq 3772	Equivalence class used to construct positive fractions.
class ~Q
Syntax	cnq 3773	Set of positive fractions.
class Q
Syntax	c1q 3774	The positive fraction constant 1.
class 1Q
Syntax	cplq 3775	Positive fraction addition.
class +Q
Syntax	cmq 3776	Positive fraction multiplication.
class ·Q
Syntax	crq 3777	Positive fraction reciprocal operation.
class *Q
Syntax	cltq 3778	Positive fraction ordering relation.
class <Q
Syntax	cnp 3779	Set of positive reals.
class P
Syntax	c1p 3780	Positive real constant 1.
class 1P
Syntax	cpp 3781	Positive real addition.
class +P
Syntax	cmp 3782	Positive real multiplication.
class ·P
Syntax	cltp 3783	Positive real ordering relation.
class <P
Syntax	cplpr 3784	Signed real pre-addition.
class +pR
Syntax	cmpr 3785	Signed real pre-multiplication.
class ·pR
Syntax	cer 3786	Equivalence class used to construct signed reals.
class ~R
Syntax	cnr 3787	Set of signed reals.
class R
Syntax	c0r 3788	The signed real constant 0.
class 0R
Syntax	c1r 3789	The signed real constant 1.
class 1R
Syntax	cm1r 3790	The signed real constant -1.
class -1R
Syntax	cplr 3791	Signed real addition.
class +R
Syntax	cmr 3792	Signed real multiplication.
class ·R
Syntax	cltr 3793	Signed real ordering relation.
class <R
Definition	df-ni 3794	Define the class of positive integers. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction.
⊢ N = (ω ∖ {∅})
Definition	df-pli 3795	Define addition on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction.
⊢ +N = ( +o ↾ (N × N))
Definition	df-mi 3796	Define multiplication on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction.
⊢ ·N = ( ·o ↾ (N × N))
Definition	df-lti 3797	Define 'less than' on positive integers. This is a "temporary" set used in the construction of complex numbers df-c 4034, and is intended to be used only by the construction.
⊢ <N = (E ∩ (N × N))
Theorem	elni 3798	Membership in the class of positive integers.
⊢ (A ∈ N ↔ (A ∈ ω ∧ ¬ A = ∅))
Theorem	elni2 3799	Membership in the class of positive integers.
⊢ (A ∈ N ↔ (A ∈ ω ∧ ∅ ∈ A))
Theorem	pinn 3800	A positive integer is a natural number.
⊢ (A ∈ N → A ∈ ω)