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Theorem bren2 8138
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 8134 . . 3 (𝐴𝐵𝐴𝐵)
2 sdomnen 8136 . . . 4 (𝐴𝐵 → ¬ 𝐴𝐵)
32con2i 136 . . 3 (𝐴𝐵 → ¬ 𝐴𝐵)
41, 3jca 501 . 2 (𝐴𝐵 → (𝐴𝐵 ∧ ¬ 𝐴𝐵))
5 brdom2 8137 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
65biimpi 206 . . 3 (𝐴𝐵 → (𝐴𝐵𝐴𝐵))
76orcanai 987 . 2 ((𝐴𝐵 ∧ ¬ 𝐴𝐵) → 𝐴𝐵)
84, 7impbii 199 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 382  wo 836   class class class wbr 4786  cen 8104  cdom 8105  csdm 8106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-f1o 6036  df-en 8108  df-dom 8109  df-sdom 8110
This theorem is referenced by:  marypha1lem  8493  tskwe  8974  infxpenlem  9034  cdainflem  9213  axcclem  9479  alephsuc3  9602  gchen1  9647  gchen2  9648  inatsk  9800  ufilen  21947  dirith2  25431  f1ocnt  29892  lindsenlbs  33730  mblfinlem1  33772  axccdom  39927  axccd2  39941
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