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Theorem bren2 8979
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 8975 . . 3 (𝐴𝐵𝐴𝐵)
2 sdomnen 8977 . . . 4 (𝐴𝐵 → ¬ 𝐴𝐵)
32con2i 139 . . 3 (𝐴𝐵 → ¬ 𝐴𝐵)
41, 3jca 513 . 2 (𝐴𝐵 → (𝐴𝐵 ∧ ¬ 𝐴𝐵))
5 brdom2 8978 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
65biimpi 215 . . 3 (𝐴𝐵 → (𝐴𝐵𝐴𝐵))
76orcanai 1002 . 2 ((𝐴𝐵 ∧ ¬ 𝐴𝐵) → 𝐴𝐵)
84, 7impbii 208 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397  wo 846   class class class wbr 5149  cen 8936  cdom 8937  csdm 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-f1o 6551  df-en 8940  df-dom 8941  df-sdom 8942
This theorem is referenced by:  marypha1lem  9428  tskwe  9945  infxpenlem  10008  cdainflem  10182  axcclem  10452  alephsuc3  10575  gchen1  10620  gchen2  10621  inatsk  10773  ufilen  23434  dirith2  27031  f1ocnt  32013  lindsenlbs  36483  mblfinlem1  36525  axccdom  43921  axccd2  43929
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