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Theorem isfi 8021
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 8001 . . 3 Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥}
21eleq2i 2722 . 2 (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥})
3 relen 8002 . . . . 5 Rel ≈
43brrelexi 5192 . . . 4 (𝐴𝑥𝐴 ∈ V)
54rexlimivw 3058 . . 3 (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ V)
6 breq1 4688 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
76rexbidv 3081 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦𝑥 ↔ ∃𝑥 ∈ ω 𝐴𝑥))
85, 7elab3 3390 . 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 1523  wcel 2030  {cab 2637  wrex 2942  Vcvv 3231   class class class wbr 4685  ωcom 7107  cen 7994  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-en 7998  df-fin 8001
This theorem is referenced by:  snfi  8079  php3  8187  onfin  8192  ominf  8213  isinf  8214  enfi  8217  ssnnfi  8220  ssfi  8221  dif1en  8234  findcard  8240  findcard2  8241  findcard3  8244  nnsdomg  8260  isfiniteg  8261  unfi  8268  fiint  8278  pwfi  8302  finnum  8812  ficardom  8825  dif1card  8871  infpwfien  8923  ficard  9425  hashkf  13159  finminlem  32437
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