Theorem isfi 6912

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 6890	. . 3 |- Fin = { y | E. x e. om y ~~ x }
2	1	eleq2i 2296	. 2 |- ( A e. Fin <-> A e. { y | E. x e. om y ~~ x } )
3		relen 6891	. . . . 5 |- Rel ~~
4	3	brrelex1i 4762	. . . 4 |- ( A ~~ x -> A e. _V )
5	4	rexlimivw 2644	. . 3 |- ( E. x e. om A ~~ x -> A e. _V )
6		breq1 4086	. . . 4 |- ( y = A -> ( y ~~ x <-> A ~~ x ) )
7	6	rexbidv 2531	. . 3 |- ( y = A -> ( E. x e. om y ~~ x <-> E. x e. om A ~~ x ) )
8	5, 7	elab3 2955	. 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 1395   e. wcel 2200   {cab 2215   E.wrex 2509   _Vcvv 2799   class class class wbr 4083   omcom 4682   ~~ cen 6885   Fincfn 6887

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-en 6888  df-fin 6890

This theorem is referenced by:  snfig  6967  fict  7030  fidceq  7031  nnfi  7034  enfi  7035  ssfilem  7037  dif1enen  7042  php5fin  7044  fisbth  7045  fin0  7047  fin0or  7048  diffitest  7049  findcard  7050  findcard2  7051  findcard2s  7052  diffisn  7055  infnfi  7057  fientri3  7077  unsnfi  7081  unsnfidcex  7082  unsnfidcel  7083  fiintim  7093  fidcenumlemim  7119  finnum  7355  ficardon  7361  hashcl  11003  hashen  11006  fihashdom  11025  hashun  11027  zfz1iso  11063