Theorem kardex 8930

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 3059	. . 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 3343	. . . . . 6 ⊢ 𝑥 ∈ V
3		breq1 4807	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑧 ≈ 𝐴 ↔ 𝑥 ≈ 𝐴))
4	2, 3	elab 3490	. . . . 5 ⊢ (𝑥 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} ↔ 𝑥 ≈ 𝐴)
5		breq1 4807	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝑧 ≈ 𝐴 ↔ 𝑦 ≈ 𝐴))
6	5	ralab 3508	. . . . 5 ⊢ (∀𝑦 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
7	4, 6	anbi12i 735	. . . 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 2877	. . 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 2782	. 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 8921	. 2 ⊢ {𝑥 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} ∣ ∀𝑦 ∈ {𝑧 ∣ 𝑧 ≈ 𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
11	9, 10	eqeltrri 2836	1 ⊢ {𝑥 ∣ (𝑥 ≈ 𝐴 ∧ ∀𝑦(𝑦 ≈ 𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1630   ∈ wcel 2139  {cab 2746  ∀wral 3050  {crab 3054  Vcvv 3340   ⊆ wss 3715   class class class wbr 4804  ‘cfv 6049   ≈ cen 8118  rankcrnk 8799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-reg 8662  ax-inf2 8711

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-r1 8800  df-rank 8801

This theorem is referenced by: (None)