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Theorem bren2 7930
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 7926 . . 3 (𝐴𝐵𝐴𝐵)
2 sdomnen 7928 . . . 4 (𝐴𝐵 → ¬ 𝐴𝐵)
32con2i 134 . . 3 (𝐴𝐵 → ¬ 𝐴𝐵)
41, 3jca 554 . 2 (𝐴𝐵 → (𝐴𝐵 ∧ ¬ 𝐴𝐵))
5 brdom2 7929 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
65biimpi 206 . . 3 (𝐴𝐵 → (𝐴𝐵𝐴𝐵))
76orcanai 951 . 2 ((𝐴𝐵 ∧ ¬ 𝐴𝐵) → 𝐴𝐵)
84, 7impbii 199 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 383  wa 384   class class class wbr 4613  cen 7896  cdom 7897  csdm 7898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-f1o 5854  df-en 7900  df-dom 7901  df-sdom 7902
This theorem is referenced by:  marypha1lem  8283  tskwe  8720  infxpenlem  8780  cdainflem  8957  axcclem  9223  alephsuc3  9346  gchen1  9391  gchen2  9392  inatsk  9544  ufilen  21644  dirith2  25117  f1ocnt  29400  lindsenlbs  33036  mblfinlem1  33078  axccdom  38890  axccd2  38904
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