Theorem eqeng 7989

Description: Equality implies equinumerosity. (Contributed by NM, 26-Oct-2003.)

Assertion

Ref	Expression
eqeng	|- ( A e. V -> ( A = B -> A ~~ B ) )

Proof of Theorem eqeng

Step	Hyp	Ref	Expression
1		enrefg 7987	. 2 |- ( A e. V -> A ~~ A )
2		breq2 4657	. 2 |- ( A = B -> ( A ~~ A <-> A ~~ B ) )
3	1, 2	syl5ibcom 235	1 |- ( A e. V -> ( A = B -> A ~~ B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   class class class wbr 4653   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-en 7956

This theorem is referenced by:  idssen  8000  nneneq  8143  onomeneq  8150  pr2ne  8828  alephord  8898  alephdom  8904  fin23lem25  9146  alephadd  9399  rp-isfinite5  37863