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Theorem isfi 6177
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 6160 . . 3 Fin = {yx 𝜔 yx}
21eleq2i 2101 . 2 (A Fin ↔ A {yx 𝜔 yx})
3 relen 6161 . . . . 5 Rel ≈
43brrelexi 4327 . . . 4 (AxA V)
54rexlimivw 2423 . . 3 (x 𝜔 AxA V)
6 breq1 3758 . . . 4 (y = A → (yxAx))
76rexbidv 2321 . . 3 (y = A → (x 𝜔 yxx 𝜔 Ax))
85, 7elab3 2688 . 2 (A {yx 𝜔 yx} ↔ x 𝜔 Ax)
92, 8bitri 173 1 (A Fin ↔ x 𝜔 Ax)
Colors of variables: wff set class
Syntax hints:  wb 98   = wceq 1242   wcel 1390  {cab 2023  wrex 2301  Vcvv 2551   class class class wbr 3755  𝜔com 4256  cen 6155  Fincfn 6157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-en 6158  df-fin 6160
This theorem is referenced by:  snfig  6227  nnfi  6251  enfi  6252  ssfiexmid  6254
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