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Theorem kardex 9041
 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 3126 . . 3 {𝑥 ∈ {𝑧𝑧𝐴} ∣ ∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝑥 ∈ {𝑧𝑧𝐴} ∧ ∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦))}
2 vex 3417 . . . . . 6 𝑥 ∈ V
3 breq1 4878 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝐴𝑥𝐴))
42, 3elab 3571 . . . . 5 (𝑥 ∈ {𝑧𝑧𝐴} ↔ 𝑥𝐴)
5 breq1 4878 . . . . . 6 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
65ralab 3590 . . . . 5 (∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦) ↔ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))
74, 6anbi12i 620 . . . 4 ((𝑥 ∈ {𝑧𝑧𝐴} ∧ ∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)) ↔ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦))))
87abbii 2944 . . 3 {𝑥 ∣ (𝑥 ∈ {𝑧𝑧𝐴} ∧ ∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦))} = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
91, 8eqtri 2849 . 2 {𝑥 ∈ {𝑧𝑧𝐴} ∣ ∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} = {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))}
10 scottex 9032 . 2 {𝑥 ∈ {𝑧𝑧𝐴} ∣ ∀𝑦 ∈ {𝑧𝑧𝐴} (rank‘𝑥) ⊆ (rank‘𝑦)} ∈ V
119, 10eqeltrri 2903 1 {𝑥 ∣ (𝑥𝐴 ∧ ∀𝑦(𝑦𝐴 → (rank‘𝑥) ⊆ (rank‘𝑦)))} ∈ V
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1654   ∈ wcel 2164  {cab 2811  ∀wral 3117  {crab 3121  Vcvv 3414   ⊆ wss 3798   class class class wbr 4875  ‘cfv 6127   ≈ cen 8225  rankcrnk 8910 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-reg 8773  ax-inf2 8822 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-tp 4404  df-op 4406  df-uni 4661  df-int 4700  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-tr 4978  df-id 5252  df-eprel 5257  df-po 5265  df-so 5266  df-fr 5305  df-we 5307  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-pred 5924  df-ord 5970  df-on 5971  df-lim 5972  df-suc 5973  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-om 7332  df-wrecs 7677  df-recs 7739  df-rdg 7777  df-r1 8911  df-rank 8912 This theorem is referenced by: (None)
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