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Theorem isfi 6666
 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 6648 . . 3 Fin = {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥}
21eleq2i 2207 . 2 (𝐴 ∈ Fin ↔ 𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥})
3 relen 6649 . . . . 5 Rel ≈
43brrelex1i 4593 . . . 4 (𝐴𝑥𝐴 ∈ V)
54rexlimivw 2549 . . 3 (∃𝑥 ∈ ω 𝐴𝑥𝐴 ∈ V)
6 breq1 3941 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
76rexbidv 2440 . . 3 (𝑦 = 𝐴 → (∃𝑥 ∈ ω 𝑦𝑥 ↔ ∃𝑥 ∈ ω 𝐴𝑥))
85, 7elab3 2841 . 2 (𝐴 ∈ {𝑦 ∣ ∃𝑥 ∈ ω 𝑦𝑥} ↔ ∃𝑥 ∈ ω 𝐴𝑥)
92, 8bitri 183 1 (𝐴 ∈ Fin ↔ ∃𝑥 ∈ ω 𝐴𝑥)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104   = wceq 1332   ∈ wcel 1481  {cab 2126  ∃wrex 2418  Vcvv 2690   class class class wbr 3938  ωcom 4514   ≈ cen 6643  Fincfn 6645 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4141 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-br 3939  df-opab 3999  df-xp 4556  df-rel 4557  df-en 6646  df-fin 6648 This theorem is referenced by:  snfig  6719  fict  6773  fidceq  6774  nnfi  6777  enfi  6778  ssfilem  6780  dif1enen  6785  php5fin  6787  fisbth  6788  fin0  6790  fin0or  6791  diffitest  6792  findcard  6793  findcard2  6794  findcard2s  6795  diffisn  6798  infnfi  6800  fientri3  6819  unsnfi  6823  unsnfidcex  6824  unsnfidcel  6825  fiintim  6833  fidcenumlemim  6856  finnum  7066  hashcl  10586  hashen  10589  fihashdom  10608  hashun  10610  zfz1iso  10643
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