Theorem kardex 8925

Description: The collection of all sets equinumerous to a set 𝐴 and having the least possible rank is a set. This is the part of the justification of the definition of kard of [Enderton] p. 222. (Contributed by NM, 14-Dec-2003.)

Assertion

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

Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem kardex

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rab 3070	. . 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		vex 3354	. . . . . 6 ⊢ 𝑥 ∈ V
3		breq1 4790	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴))
4	2, 3	elab 3501	. . . . 5 ⊢ (𝑥 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} ↔ 𝑥 ≈ 𝐴)
5		breq1 4790	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
6	5	ralab 3519	. . . . 5 ⊢ (∀𝑦 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
7	4, 6	anbi12i 612	. . . 4 ⊢ ((𝑥 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} ∧ ∀𝑦 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))))
8	7	abbii 2888	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} ∧ ∀𝑦 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} (rank‘𝑥) ⊆ (rank‘𝑦))} = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
9	1, 8	eqtri 2793	. 2 ⊢ {𝑥 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
10		scottex 8916	. 2 ⊢ {𝑥 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
11	9, 10	eqeltrri 2847	1 ⊢ {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V

Syntax hints:   → wi 4   ∧ wa 382  ∀wal 1629   ∈ wcel 2145  {cab 2757  ∀wral 3061  {crab 3065  Vcvv 3351   ⊆ wss 3723   class class class wbr 4787  ‘cfv 6030   ≈ cen 8110  rankcrnk 8794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100  ax-reg 8657  ax-inf2 8706

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-int 4613  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-om 7217  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-r1 8795  df-rank 8796

This theorem is referenced by: (None)