Theorem kardex 9913

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 3421	. . 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 3468	. . . . . 6 ⊢ 𝑥 ∈ V
3		breq1 5127	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴))
4	2, 3	elab 3663	. . . . 5 ⊢ (𝑥 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} ↔ 𝑥 ≈ 𝐴)
5		breq1 5127	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
6	5	ralab 3681	. . . . 5 ⊢ (∀𝑦 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
7	4, 6	anbi12i 628	. . . 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 2803	. . 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 2759	. 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 9904	. 2 ⊢ {𝑥 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
11	9, 10	eqeltrri 2832	1 ⊢ {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   ∈ wcel 2109  {cab 2714  ∀wral 3052  {crab 3420  Vcvv 3464   ⊆ wss 3931   class class class wbr 5124  ‘cfv 6536   ≈ cen 8961  rankcrnk 9782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-reg 9611  ax-inf2 9660

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7867  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-r1 9783  df-rank 9784

This theorem is referenced by: (None)