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Theorem isfi 6760
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 6742 . . 3 Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥}
21eleq2i 2244 . 2 (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥})
3 relen 6743 . . . . 5 Rel ≈
43brrelex1i 4669 . . . 4 (𝐴𝑥𝐴 ∈ V)
54rexlimivw 2590 . . 3 (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ V)
6 breq1 4006 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
76rexbidv 2478 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦𝑥 ↔ ∃𝑥 ∈ ω 𝐴𝑥))
85, 7elab3 2889 . 2 (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥} ↔ ∃𝑥 ∈ ω 𝐴𝑥)
92, 8bitri 184 1 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353  wcel 2148  {cab 2163  wrex 2456  Vcvv 2737   class class class wbr 4003  ωcom 4589  cen 6737  Fincfn 6739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-en 6740  df-fin 6742
This theorem is referenced by:  snfig  6813  fict  6867  fidceq  6868  nnfi  6871  enfi  6872  ssfilem  6874  dif1enen  6879  php5fin  6881  fisbth  6882  fin0  6884  fin0or  6885  diffitest  6886  findcard  6887  findcard2  6888  findcard2s  6889  diffisn  6892  infnfi  6894  fientri3  6913  unsnfi  6917  unsnfidcex  6918  unsnfidcel  6919  fiintim  6927  fidcenumlemim  6950  finnum  7181  hashcl  10756  hashen  10759  fihashdom  10778  hashun  10780  zfz1iso  10816
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