MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brdom Unicode version

Theorem brdom 6874
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 6872 . 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 1528    e. wcel 1684   _Vcvv 2788   class class class wbr 4023   -1-1->wf1 5252    ~<_ cdom 6861
This theorem is referenced by:  domen  6875  domtr  6914  sbthlem10  6980  1sdom  7065  ac10ct  7661  domtriomlem  8068  2ndcdisj  17182  birthdaylem3  20248
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-fn 5258  df-f 5259  df-f1 5260  df-dom 6865
  Copyright terms: Public domain W3C validator