Theorem kardex 3550

Description: The collection of all sets equinumerous to a set A and having least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222.

Hypothesis

Ref	Expression
kardex.1	⊢ A ∈ V

Assertion

Ref	Expression
kardex	⊢ {x∣(x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)))} ∈ V

Distinct variable group(s):   x,y,A

Proof of Theorem kardex

Step	Hyp	Ref	Expression
1		df-rab 1208	. . 3 ⊢ {x ∈ {z∣z ≈ A}∣∀y ∈ {z∣z ≈ A} (rank ‘x) ⊆ (rank ‘y)} = {x∣(x ∈ {z∣z ≈ A} ∧ ∀y ∈ {z∣z ≈ A} (rank ‘x) ⊆ (rank ‘y))}
2		visset 1350	. . . . . 6 ⊢ x ∈ V
3		breq1 2065	. . . . . 6 ⊢ (z = x → (z ≈ A ↔ x ≈ A))
4	2, 3	elab 1415	. . . . 5 ⊢ (x ∈ {z∣z ≈ A} ↔ x ≈ A)
5		df-ral 1205	. . . . . 6 ⊢ (∀y ∈ {z∣z ≈ A} (rank ‘x) ⊆ (rank ‘y) ↔ ∀y(y ∈ {z∣z ≈ A} → (rank ‘x) ⊆ (rank ‘y)))
6		visset 1350	. . . . . . . . 9 ⊢ y ∈ V
7		breq1 2065	. . . . . . . . 9 ⊢ (z = y → (z ≈ A ↔ y ≈ A))
8	6, 7	elab 1415	. . . . . . . 8 ⊢ (y ∈ {z∣z ≈ A} ↔ y ≈ A)
9	8	imbi1i 161	. . . . . . 7 ⊢ ((y ∈ {z∣z ≈ A} → (rank ‘x) ⊆ (rank ‘y)) ↔ (y ≈ A → (rank ‘x) ⊆ (rank ‘y)))
10	9	bial 695	. . . . . 6 ⊢ (∀y(y ∈ {z∣z ≈ A} → (rank ‘x) ⊆ (rank ‘y)) ↔ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)))
11	5, 10	bitr 151	. . . . 5 ⊢ (∀y ∈ {z∣z ≈ A} (rank ‘x) ⊆ (rank ‘y) ↔ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)))
12	4, 11	anbi12i 369	. . . 4 ⊢ ((x ∈ {z∣z ≈ A} ∧ ∀y ∈ {z∣z ≈ A} (rank ‘x) ⊆ (rank ‘y)) ↔ (x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y))))
13	12	biabi 1181	. . 3 ⊢ {x∣(x ∈ {z∣z ≈ A} ∧ ∀y ∈ {z∣z ≈ A} (rank ‘x) ⊆ (rank ‘y))} = {x∣(x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)))}
14	1, 13	eqtr 1119	. 2 ⊢ {x ∈ {z∣z ≈ A}∣∀y ∈ {z∣z ≈ A} (rank ‘x) ⊆ (rank ‘y)} = {x∣(x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)))}
15		scottex 3541	. 2 ⊢ {x ∈ {z∣z ≈ A}∣∀y ∈ {z∣z ≈ A} (rank ‘x) ⊆ (rank ‘y)} ∈ V
16	14, 15	eqeltrr 1160	1 ⊢ {x∣(x ≈ A ∧ ∀y(y ≈ A → (rank ‘x) ⊆ (rank ‘y)))} ∈ V

Syntax hints:   → wi 2   ∧ wa 196  ∀wal 672  {cab 1090   ∈ wcel 1092  ∀wral 1201  {crab 1204  Vcvv 1348   ⊆ wss 1487   class class class wbr 2054   ‘cfv 2422   ≈ cen 3271  rankcrnk 3486

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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