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

Theorem domen 6875
 Description: Dominance in terms of equinumerosity. Example 1 of [Enderton] p. 146. (Contributed by NM, 15-Jun-1998.)
Hypothesis
Ref Expression
bren.1
Assertion
Ref Expression
domen
Distinct variable groups:   ,   ,

Proof of Theorem domen
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 bren.1 . . 3
21brdom 6874 . 2
3 vex 2791 . . . . . 6
43f11o 5506 . . . . 5
54exbii 1569 . . . 4
6 excom 1786 . . . 4
75, 6bitri 240 . . 3
8 bren 6871 . . . . . 6
98anbi1i 676 . . . . 5
10 19.41v 1842 . . . . 5
119, 10bitr4i 243 . . . 4
1211exbii 1569 . . 3
137, 12bitr4i 243 . 2
142, 13bitri 240 1
 Colors of variables: wff set class Syntax hints:   wb 176   wa 358  wex 1528   wcel 1684  cvv 2788   wss 3152   class class class wbr 4023  wf1 5252  wf1o 5254   cen 6860   cdom 6861 This theorem is referenced by:  domeng  6876  infcntss  7130  cdainf  7818  ramub2  13061  ram0  13069 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-fo 5261  df-f1o 5262  df-en 6864  df-dom 6865
 Copyright terms: Public domain W3C validator