Theorem enrefg 7003

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 5654	. . 3 |- ( _I |` A ) : A -1-1-onto-> A
2		f1oen2g 6994	. . 3 |- ( ( A e. V /\ A e. V /\ ( _I |` A ) : A -1-1-onto-> A ) -> A ~~ A )
3	1, 2	mp3an3 1363	. 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 2203   class class class wbr 4109   _I cid 4409   |` cres 4751   -1-1-onto->wf1o 5351   ~~ cen 6973

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-en 6976

This theorem is referenced by:  enref  7004  eqeng  7005  domrefg  7006  mapdom1g  7100  fidifsnen  7125  nnfi  7127  onenon  7480  oncardval  7482  cardonle  7483  dju1en  7520  xpdjuen  7525  iseqf1olemqf1o  10868  hashun  11169  lgseisenlem2  15944