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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 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
Distinct variable group:   𝑥,𝑦,𝐴

Proof of Theorem kardex
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 df-rab 3070 . . 3 {𝑥 ∈ {𝑧𝑧𝐴} ∣ ∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝑥 ∈ {𝑧𝑧𝐴} ∧ ∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦))}
2 vex 3354 . . . . . 6 𝑥 ∈ V
3 breq1 4790 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
42, 3elab 3501 . . . . 5 (𝑥 ∈ {𝑧𝑧𝐴} ↔ 𝑥𝐴)
5 breq1 4790 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
65ralab 3519 . . . . 5 (∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
74, 6anbi12i 612 . . . 4 ((𝑥 ∈ {𝑧𝑧𝐴} ∧ ∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))))
87abbii 2888 . . 3 {𝑥 ∣ (𝑥 ∈ {𝑧𝑧𝐴} ∧ ∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦))} = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
91, 8eqtri 2793 . 2 {𝑥 ∈ {𝑧𝑧𝐴} ∣ ∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
10 scottex 8916 . 2 {𝑥 ∈ {𝑧𝑧𝐴} ∣ ∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
119, 10eqeltrri 2847 1 {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
Colors of variables: wff setvar class
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)
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