Metamath Proof Explorer - mmtheorems37

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Statement List for Metamath Proof Explorer - 3601-3700 - Page 37 of 58

Type	Label	Description
Statement
Theorem	numthcor 3601	Any set is strictly dominated by some ordinal.
|- (A e. B -> E.x e. On A ~< x)
Theorem	weth 3602	Well-ordering theorem: any set A can be well-ordered. This is an equivalent of the Axiom of Choice. Theorem 6 of [Suppes] p. 242.
|- A e. V   =>   |- E.x x We A
Theorem	zornlem1 3603	Lemma for Zorn's lemma.
|- A e. V   &   |- B = {f | E.h e. On (f Fn h /\ A.t e. h (f` t) = (G` (f |` t)))}   &   |- F = U.B   &   |- C = {z e. A | A.g e. ran fgRz}   &   |- D = {z e. A | A.g e. (F"x)gRz}   &   |- G = {<.f, t>. | t = U.{v e. C | A.u e. C -. uwv}}   =>   |- ((x e. On /\ (w We A /\ -. D = (/))) -> (F` x) e. D)
Theorem	zornlem2 3604	Lemma for Zorn's lemma.
|- A e. V   &   |- B = {f | E.h e. On (f Fn h /\ A.t e. h (f` t) = (G` (f |` t)))}   &   |- F = U.B   &   |- C = {z e. A | A.g e. ran fgRz}   &   |- D = {z e. A | A.g e. (F"x)gRz}   &   |- G = {<.f, t>. | t = U.{v e. C | A.u e. C -. uwv}}   =>   |- ((x e. On /\ (w We A /\ -. D = (/))) -> (y e. x -> (F` y)R(F` x)))
Theorem	zornlem3 3605	Lemma for Zorn's lemma.
|- A e. V   &   |- B = {f | E.h e. On (f Fn h /\ A.t e. h (f` t) = (G` (f |` t)))}   &   |- F = U.B   &   |- C = {z e. A | A.g e. ran fgRz}   &   |- D = {z e. A | A.g e. (F"x)gRz}   &   |- G = {<.f, t>. | t = U.{v e. C | A.u e. C -. uwv}}   =>   |- ((R Po A /\ (x e. On /\ (w We A /\ -. D = (/)))) -> (y e. x -> -. (F` x) = (F` y)))
Theorem	zornlem4 3606	Lemma for Zorn's lemma.
|- A e. V   &   |- B = {f | E.h e. On (f Fn h /\ A.t e. h (f` t) = (G` (f |` t)))}   &   |- F = U.B   &   |- C = {z e. A | A.g e. ran fgRz}   &   |- D = {z e. A | A.g e. (F"x)gRz}   &   |- G = {<.f, t>. | t = U.{v e. C | A.u e. C -. uwv}}   =>   |- ((R Po A /\ w We A) -> E.x e. On D = (/))
Theorem	zornlem5 3607	Lemma for Zorn's lemma.
|- A e. V   &   |- B = {f | E.h e. On (f Fn h /\ A.t e. h (f` t) = (G` (f |` t)))}   &   |- F = U.B   &   |- C = {z e. A | A.g e. ran fgRz}   &   |- D = {z e. A | A.g e. (F"x)gRz}   &   |- G = {<.f, t>. | t = U.{v e. C | A.u e. C -. uwv}}   &   |- H = {z e. A | A.g e. (F"y)gRz}   =>   |- (((w We A /\ x e. On) /\ A.y e. x -. H = (/)) -> (F"x) (_ A)
Theorem	zornlem6 3608	Lemma for Zorn's lemma.
|- A e. V   &   |- B = {f | E.h e. On (f Fn h /\ A.t e. h (f` t) = (G` (f |` t)))}   &   |- F = U.B   &   |- C = {z e. A | A.g e. ran fgRz}   &   |- D = {z e. A | A.g e. (F"x)gRz}   &   |- G = {<.f, t>. | t = U.{v e. C | A.u e. C -. uwv}}   &   |- H = {z e. A | A.g e. (F"y)gRz}   =>   |- (R Po A -> (((w We A /\ x e. On) /\ A.y e. x -. H = (/)) -> R Or (F"x)))
Theorem	zornlem7 3609	Lemma for Zorn's lemma.
|- A e. V   &   |- B = {f | E.h e. On (f Fn h /\ A.t e. h (f` t) = (G` (f |` t)))}   &   |- F = U.B   &   |- C = {z e. A | A.g e. ran fgRz}   &   |- D = {z e. A | A.g e. (F"x)gRz}   &   |- G = {<.f, t>. | t = U.{v e. C | A.u e. C -. uwv}}   &   |- H = {z e. A | A.g e. (F"y)gRz}   =>   |- ((R Po A /\ A.s((s (_ A /\ R Or s) -> E.a e. A A.r e. s (rRa \/ r = a))) -> E.a e. A A.b e. A -. aRb)
Theorem	zorn2lem 3610	Lemma for zorn2 3612.
|- (z{<.x, y>. | x (. y}w <-> z (. w)
Theorem	zorn 3611	Zorn's Lemma of [Monk1] p. 117. This theorem is equivalent to the Axiom of Choice and states that every partially ordered set A (with an ordering relation R) in which every totally ordered subset has an upper bound, contains at least one maximal element. The main proof consists of lemmas zornlem1 3603 through zornlem7 3609; this final piece mainly changes bound variables to eliminate the hypotheses of zornlem7 3609.
|- A e. V   =>   |- ((R Po A /\ A.w((w (_ A /\ R Or w) -> E.x e. A A.z e. w (zRx \/ z = x))) -> E.x e. A A.y e. A -. xRy)
Theorem	zorn2 3612	Zorn's Lemma. If the union of every chain (with respect to inclusion) in a set belongs to the set, then the set contains a maximal element. This theorem is equivalent to the Axiom of Choice. Theorem 6M of [Enderton] p. 151.
|- A e. V   =>   |- (A.z((z (_ A /\ A.x e. z A.y e. z (x (_ y \/ y (_ x)) -> U.z e. A) -> E.x e. A A.y e. A -. x (. y)
Theorem	fodom 3613	An onto function implies dominance of range over domain. Lemma 10.20 of [Kunen] p. 30. This theorem uses the Axiom of Choice ac7g 3570.
|- A e. V   =>   |- (F:A-onto->B -> B ~<_ A)
Theorem	fodomg 3614	An onto function implies dominance of range over domain.
|- (A e. C -> (F:A-onto->B -> B ~<_ A))
Theorem	fodomb 3615	Equivalence of an onto mapping and dominance for a non-empty set. Proposition 10.35 of [TakeutiZaring] p. 93.
|- A e. V   &   |- B e. V   =>   |- ((-. A = (/) /\ E.f f:A-onto->B) <-> ((/) ~< B /\ B ~<_ A))
Theorem	imadomg 3616	An image of a function under a set is dominated by the set. Proposition 10.34 of [TakeutiZaring] p. 92.
|- (A e. B -> (Fun F -> (F"A) ~<_ A))
Theorem	fnrndomg 3617	The range of a function is dominated by its domain.
|- (A e. B -> (F Fn A -> ran F ~<_ A))
Theorem	htalem 3618	Lemma for defining an emulation of Hilbert's epsilon. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem is equivalent to Hilbert's "transfinite axiom," described on that page, with the additional R We A antecedent. The element B is the epsilon that the theorem emulates.
|- A e. V   &   |- B = U.{x e. A | A.y e. A -. yRx}   =>   |- ((R We A /\ -. A = (/)) -> B e. A)
Theorem	hta 3619	A ZFC emulation of Hilbert's transfinite axiom. The set B has the properties of Hilbert's epsilon, except that it also depends on a well-ordering R. This theorem arose from discussions with Raph Levien on 5-Mar-04 about translating the HOL proof language, which uses Hilbert's epsilon. Hilbert's epsilon is described at http://plato.stanford.edu/entries/epsilon-calculus/. This theorem differs from Hilbert's transfinite axiom described on that page in that it requires R We A as an antecedent. Class A collects the sets of least rank for which ph(x) is true. Class B, which emulates the epsilon, is the minimum element in a well-ordering R on A. If a well-ordering R on A can be expressed in a closed form, as might be the case if we are working with say natural numbers, we can eliminate the antecedent with modus ponens, giving us the exact equivalent of Hilbert's transfinite axiom. Otherwise, we replace R with a dummy set variable, say w, and attach w We A as an antecedent in each step of the ZFC version of the HOL proof until the epsilon is eliminated. At that point, B (which will have w as a free variable) will no longer be present, and we can eliminate w We A by applying 19.23aiv 952 and weth 3602, using scottexs 3543 to establish the existence of A. For a version of this theorem scheme using class (meta)variables instead of wff (meta)variables, see htalem 3618.
|- A = {x | (ph /\ A.y([y / x]ph -> (rank` x) (_ (rank` y)))}   &   |- B = U.{x e. A | A.y e. A -. yRx}   =>   |- (R We A -> (ph -> [B / x]ph))
Syntax	ccrd 3620	Extend class definition to include the cardinal size function.
class card
Syntax	cale 3621	Extend class definition to include the aleph function.
class aleph
Syntax	ccf 3622	Extend class definition to include the cofinality function.
class cf
Definition	df-card 3623	Define the cardinal number function. The cardinal number of a set is the least ordinal number equinumerous to it. In other words, it is the "size" of the set. Definition of [Enderton] p. 197. See cardval 3633 for its value, cardval2 3661 for a simpler version of its value. The principle theorem relating cardinality to equinumerosity is carden 3638. Our notation is from Enderton. Other textbooks often use a double bar over the set to express this function.
|- card = {<.x, y>. | y = |^|{z e. On | z ~~ x}}
Definition	df-aleph 3624	Define the aleph function. Our definition expresses Definition 12 of [Suppes] p. 229 in a closed form, from which we derive the recursive definition as theorems aleph0 3669, alephsuc 3672, and alephlim 3670. The aleph function provides a one-to-one, onto mapping from the ordinal numbers to the infinite cardinal numbers. Roughly, any aleph is the smallest infinite cardinal number whose size is strictly greater than any aleph before it.
|- aleph = rec({<.x, y>. | y = |^|{z e. On | x ~< z}}, om)
Definition	df-cf 3625	Define the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx. See cfval 3701 for its value and a description.
|- cf = {<.x, y>. | (x e. On /\ y = |^|{z | E.w(z = (card` w) /\ (w (_ x /\ A.v e. x E.u e. w v (_ u))})}
Theorem	oncardval 3626	The value of the cardinal number function with an ordinal number as its argument. Unlike cardval 3633, this theorem does not require the Axiom of Choice.
|- (A e. On -> (card` A) = |^|{x e. On | x ~~ A})
Theorem	oncardon 3627	The cardinal number of an ordinal number is an ordinal number. Unlike cardon 3634, this theorem does not require the Axiom of Choice.
|- (A e. On -> (card` A) e. On)
Theorem	oncardid 3628	Any ordinal number is equinumerous to its cardinal number. Unlike cardid 3635, this theorem does not require the Axiom of Choice.
|- (A e. On -> (card` A) ~~ A)
Theorem	cardonle 3629	The cardinal of an ordinal number is less than or equal to the ordinal number. Proposition 10.6(3) of [TakeutiZaring] p. 85.
|- (A e. On -> (card` A) (_ A)
Theorem	card0 3630	The cardinality of the empty set is the empty set.
|- (card` (/)) = (/)
Theorem	cardnn 3631	The cardinality of a natural number is the number. Corollary 10.23 of [TakeutiZaring] p. 90.
|- (A e. om -> (card` A) = A)
Theorem	cardom 3632	The set of natural numbers is a cardinal number. Theorem 18.11 of [Monk1] p. 133.
|- (card` om) = om
Theorem	cardval 3633	The value of the cardinal number function. Definition 10.4 of [TakeutiZaring] p. 85. See cardval2 3661 for a simpler version of its value.