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Theorem brdom 7017
Description: Dominance relation. (Contributed by NM, 15-Jun-1998.)
Hypothesis
Ref Expression
bren.1  |-  B  e. 
_V
Assertion
Ref Expression
brdom  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
Distinct variable groups:    A, f    B, f

Proof of Theorem brdom
StepHypRef Expression
1 bren.1 . 2  |-  B  e. 
_V
2 brdomg 7015 . 2  |-  ( B  e.  _V  ->  ( A  ~<_  B  <->  E. f 
f : A -1-1-> B
) )
31, 2ax-mp 8 1  |-  ( A  ~<_  B  <->  E. f  f : A -1-1-> B )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1546    e. wcel 1715   _Vcvv 2873   class class class wbr 4125   -1-1->wf1 5355    ~<_ cdom 7004
This theorem is referenced by:  domen  7018  domtr  7057  sbthlem10  7123  1sdom  7208  ac10ct  7808  domtriomlem  8215  2ndcdisj  17399  birthdaylem3  20470
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-xp 4798  df-rel 4799  df-cnv 4800  df-dm 4802  df-rn 4803  df-fn 5361  df-f 5362  df-f1 5363  df-dom 7008
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