MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  bren2 Structured version   Visualization version   GIF version

Theorem bren2 9004
Description: Equinumerosity expressed in terms of dominance and strict dominance. (Contributed by NM, 23-Oct-2004.)
Assertion
Ref Expression
bren2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))

Proof of Theorem bren2
StepHypRef Expression
1 endom 9000 . . 3 (𝐴𝐵𝐴𝐵)
2 sdomnen 9002 . . . 4 (𝐴𝐵 → ¬ 𝐴𝐵)
32con2i 139 . . 3 (𝐴𝐵 → ¬ 𝐴𝐵)
41, 3jca 510 . 2 (𝐴𝐵 → (𝐴𝐵 ∧ ¬ 𝐴𝐵))
5 brdom2 9003 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
65biimpi 215 . . 3 (𝐴𝐵 → (𝐴𝐵𝐴𝐵))
76orcanai 1000 . 2 ((𝐴𝐵 ∧ ¬ 𝐴𝐵) → 𝐴𝐵)
84, 7impbii 208 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 394  wo 845   class class class wbr 5149  cen 8961  cdom 8962  csdm 8963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-rel 5685  df-f1o 6556  df-en 8965  df-dom 8966  df-sdom 8967
This theorem is referenced by:  marypha1lem  9458  tskwe  9975  infxpenlem  10038  cdainflem  10212  axcclem  10482  alephsuc3  10605  gchen1  10650  gchen2  10651  inatsk  10803  ufilen  23878  dirith2  27506  f1ocnt  32652  lindsenlbs  37219  mblfinlem1  37261  axccdom  44734  axccd2  44742
  Copyright terms: Public domain W3C validator