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Theorem isnum3 9993
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 9992 . 2 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
2 cardon 9983 . . 3 (card‘𝐴) ∈ On
3 isnumi 9985 . . 3 (((card‘𝐴) ∈ On ∧ (card‘𝐴) ≈ 𝐴) → 𝐴 ∈ dom card)
42, 3mpan 688 . 2 ((card‘𝐴) ≈ 𝐴𝐴 ∈ dom card)
51, 4impbii 208 1 (𝐴 ∈ dom card ↔ (card‘𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2098   class class class wbr 5152  dom cdm 5681  Oncon0 6375  cfv 6553  cen 8970  cardccrd 9974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5303  ax-nul 5310  ax-pr 5432
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5579  df-eprel 5585  df-po 5593  df-so 5594  df-fr 5636  df-we 5638  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-ima 5694  df-ord 6378  df-on 6379  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-en 8974  df-card 9978
This theorem is referenced by:  ttukey2g  10555
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