Theorem kardex 9307

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 3115	. . 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 3444	. . . . . 6 ⊢ 𝑥 ∈ V
3		breq1 5033	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴))
4	2, 3	elab 3615	. . . . 5 ⊢ (𝑥 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} ↔ 𝑥 ≈ 𝐴)
5		breq1 5033	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
6	5	ralab 3632	. . . . 5 ⊢ (∀𝑦 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
7	4, 6	anbi12i 629	. . . 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 2863	. . 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 2821	. 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 9298	. 2 ⊢ {𝑥 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
11	9, 10	eqeltrri 2887	1 ⊢ {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V

Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   ∈ wcel 2111  {cab 2776  ∀wral 3106  {crab 3110  Vcvv 3441   ⊆ wss 3881   class class class wbr 5030  ‘cfv 6324   ≈ cen 8489  rankcrnk 9176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-reg 9040  ax-inf2 9088

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-r1 9177  df-rank 9178

This theorem is referenced by: (None)