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Theorem isfi 6241
 Description: Express " is finite." Definition 10.29 of [TakeutiZaring] p. 91 (whose " " is a predicate instead of a class). (Contributed by NM, 22-Aug-2008.)
Assertion
Ref Expression
isfi
Distinct variable group:   ,

Proof of Theorem isfi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-fin 6224 . . 3
21eleq2i 2104 . 2
3 relen 6225 . . . . 5
43brrelexi 4384 . . . 4
54rexlimivw 2429 . . 3
6 breq1 3767 . . . 4
76rexbidv 2327 . . 3
85, 7elab3 2694 . 2
92, 8bitri 173 1
 Colors of variables: wff set class Syntax hints:   wb 98   wceq 1243   wcel 1393  cab 2026  wrex 2307  cvv 2557   class class class wbr 3764  com 4313   cen 6219  cfn 6221 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-en 6222  df-fin 6224 This theorem is referenced by:  snfig  6291  fidceq  6330  nnfi  6333  enfi  6334  ssfiexmid  6336  php5fin  6339  fisbth  6340  fin0  6342  fin0or  6343  diffitest  6344  findcard  6345  findcard2  6346  findcard2s  6347  diffisn  6350  fientri3  6353  finnum  6363
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