Theorem isfi 6875

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 e. Fin <-> E. x e. om A ~~ x )

Distinct variable group:   x, A

Proof of Theorem isfi

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fin 6853	. . 3 |- Fin = { y | E. x e. om y ~~ x }
2	1	eleq2i 2274	. 2 |- ( A e. Fin <-> A e. { y | E. x e. om y ~~ x } )
3		relen 6854	. . . . 5 |- Rel ~~
4	3	brrelex1i 4736	. . . 4 |- ( A ~~ x -> A e. _V )
5	4	rexlimivw 2621	. . 3 |- ( E. x e. om A ~~ x -> A e. _V )
6		breq1 4062	. . . 4 |- ( y = A -> ( y ~~ x <-> A ~~ x ) )
7	6	rexbidv 2509	. . 3 |- ( y = A -> ( E. x e. om y ~~ x <-> E. x e. om A ~~ x ) )
8	5, 7	elab3 2932	. 2 |- ( A e. { y | E. x e. om y ~~ x } <-> E. x e. om A ~~ x )
9	2, 8	bitri 184	1 |- ( A e. Fin <-> E. x e. om A ~~ x )

Syntax hints:   <-> wb 105   = wceq 1373   e. wcel 2178   {cab 2193   E.wrex 2487   _Vcvv 2776   class class class wbr 4059   omcom 4656   ~~ cen 6848   Fincfn 6850

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-rel 4700  df-en 6851  df-fin 6853

This theorem is referenced by:  snfig  6930  fict  6991  fidceq  6992  nnfi  6995  enfi  6996  ssfilem  6998  dif1enen  7003  php5fin  7005  fisbth  7006  fin0  7008  fin0or  7009  diffitest  7010  findcard  7011  findcard2  7012  findcard2s  7013  diffisn  7016  infnfi  7018  fientri3  7038  unsnfi  7042  unsnfidcex  7043  unsnfidcel  7044  fiintim  7054  fidcenumlemim  7080  finnum  7316  ficardon  7322  hashcl  10963  hashen  10966  fihashdom  10985  hashun  10987  zfz1iso  11023