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Theorem isfi 7923
Description: Express "𝐴 is finite." Definition 10.29 of [TakeutiZaring] p. 91 (whose "Fin " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)
Assertion
Ref Expression
isfi (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
Distinct variable group:   𝑥,𝐴

Proof of Theorem isfi
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-fin 7903 . . 3 Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥}
21eleq2i 2690 . 2 (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥})
3 relen 7904 . . . . 5 Rel ≈
43brrelexi 5118 . . . 4 (𝐴𝑥𝐴 ∈ V)
54rexlimivw 3022 . . 3 (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ V)
6 breq1 4616 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
76rexbidv 3045 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦𝑥 ↔ ∃𝑥 ∈ ω 𝐴𝑥))
85, 7elab3 3341 . 2 (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥} ↔ ∃𝑥 ∈ ω 𝐴𝑥)
92, 8bitri 264 1 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  {cab 2607  wrex 2908  Vcvv 3186   class class class wbr 4613  ωcom 7012  cen 7896  Fincfn 7899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-en 7900  df-fin 7903
This theorem is referenced by:  snfi  7982  php3  8090  onfin  8095  ominf  8116  isinf  8117  enfi  8120  ssnnfi  8123  ssfi  8124  dif1en  8137  findcard  8143  findcard2  8144  findcard3  8147  nnsdomg  8163  isfiniteg  8164  unfi  8171  fiint  8181  pwfi  8205  finnum  8718  ficardom  8731  dif1card  8777  infpwfien  8829  ficard  9331  hashkf  13059  finminlem  31954
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