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Theorem isfi 8923
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 8894 . . 3 Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥}
21eleq2i 2830 . 2 (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥})
3 relen 8895 . . . . 5 Rel ≈
43brrelex1i 5693 . . . 4 (𝐴𝑥𝐴 ∈ V)
54rexlimivw 3149 . . 3 (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ V)
6 breq1 5113 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
76rexbidv 3176 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦𝑥 ↔ ∃𝑥 ∈ ω 𝐴𝑥))
85, 7elab3 3643 . 2 (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥} ↔ ∃𝑥 ∈ ω 𝐴𝑥)
92, 8bitri 275 1 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1542  wcel 2107  {cab 2714  wrex 3074  Vcvv 3448   class class class wbr 5110  ωcom 7807  cen 8887  Fincfn 8890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5173  df-xp 5644  df-rel 5645  df-en 8891  df-fin 8894
This theorem is referenced by:  snfi  8995  findcard  9114  findcard2  9115  nnfi  9118  ssnnfi  9120  ssnnfiOLD  9121  unfi  9123  ssfiALT  9125  enfii  9140  enfiALT  9142  php3  9163  php3OLD  9175  onfin  9181  ominf  9209  ominfOLD  9210  isinf  9211  isinfOLD  9212  dif1ennnALT  9228  findcard2OLD  9235  findcard3  9236  findcard3OLD  9237  nnsdomg  9253  nnsdomgOLD  9254  isfiniteg  9255  unfiOLD  9264  fiint  9275  pwfiOLD  9298  finnum  9891  ficardom  9904  dif1card  9953  infpwfien  10005  ficard  10508  hashkf  14239  finminlem  34819  domalom  35904
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