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Theorem enfiALT 9222
Description: Shorter proof of enfi 9221 using ax-pow 5369. (Contributed by NM, 22-Aug-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
enfiALT (𝐴𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))

Proof of Theorem enfiALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 enen1 9148 . . 3 (𝐴𝐵 → (𝐴𝑥𝐵𝑥))
21rexbidv 3176 . 2 (𝐴𝐵 → (∃𝑥 ∈ ω 𝐴𝑥 ↔ ∃𝑥 ∈ ω 𝐵𝑥))
3 isfi 9003 . 2 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
4 isfi 9003 . 2 (𝐵 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐵𝑥)
52, 3, 43bitr4g 313 1 (𝐴𝐵 → (𝐴 ∈ Fin ↔ 𝐵 ∈ Fin))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wcel 2098  wrex 3067   class class class wbr 5152  ωcom 7876  cen 8967  Fincfn 8970
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 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-er 8731  df-en 8971  df-fin 8974
This theorem is referenced by: (None)
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