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Theorem isnum3 9380
 Description: A set is numerable iff it is equinumerous with its cardinal. (Contributed by Mario Carneiro, 29-Apr-2015.)
Assertion
Ref Expression
isnum3 (𝐴 ∈ dom card ↔ (card‘𝐴) ≈ 𝐴)

Proof of Theorem isnum3
StepHypRef Expression
1 cardid2 9379 . 2 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2 cardon 9370 . . 3 (card‘𝐴) ∈ On
3 isnumi 9372 . . 3 (((card‘𝐴) ∈ On ∧ (card‘𝐴) ≈ 𝐴) → 𝐴 ∈ dom card)
42, 3mpan 689 . 2 ((card‘𝐴) ≈ 𝐴𝐴 ∈ dom card)
51, 4impbii 212 1 (𝐴 ∈ dom card ↔ (card‘𝐴) ≈ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∈ wcel 2115   class class class wbr 5052  dom cdm 5542  Oncon0 6178  ‘cfv 6343   ≈ cen 8502  cardccrd 9361 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455 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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-en 8506  df-card 9365 This theorem is referenced by:  ttukey2g  9936
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