Theorem enrefg 6932

Description: Equinumerosity is reflexive. Theorem 1 of [Suppes] p. 92. (Contributed by NM, 18-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
enrefg	|- ( A e. V -> A ~~ A )

Proof of Theorem enrefg

Step	Hyp	Ref	Expression
1		f1oi 5619	. . 3 |- ( _I |` A ) : A -1-1-onto-> A
2		f1oen2g 6923	. . 3 |- ( ( A e. V /\ A e. V /\ ( _I |` A ) : A -1-1-onto-> A ) -> A ~~ A )
3	1, 2	mp3an3 1360	. 2 |- ( ( A e. V /\ A e. V ) -> A ~~ A )
4	3	anidms 397	1 |- ( A e. V -> A ~~ A )

Syntax hints:   -> wi 4   e. wcel 2200   class class class wbr 4086   _I cid 4383   |` cres 4725   -1-1-onto->wf1o 5323   ~~ cen 6902

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-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-en 6905

This theorem is referenced by:  enref  6933  eqeng  6934  domrefg  6935  mapdom1g  7028  fidifsnen  7052  nnfi  7054  onenon  7379  oncardval  7381  cardonle  7382  dju1en  7418  xpdjuen  7423  iseqf1olemqf1o  10758  hashun  11058  lgseisenlem2  15790