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Theorem bren2 8224
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 8220 . . 3 (𝐴𝐵𝐴𝐵)
2 sdomnen 8222 . . . 4 (𝐴𝐵 → ¬ 𝐴𝐵)
32con2i 137 . . 3 (𝐴𝐵 → ¬ 𝐴𝐵)
41, 3jca 508 . 2 (𝐴𝐵 → (𝐴𝐵 ∧ ¬ 𝐴𝐵))
5 brdom2 8223 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
65biimpi 208 . . 3 (𝐴𝐵 → (𝐴𝐵𝐴𝐵))
76orcanai 1026 . 2 ((𝐴𝐵 ∧ ¬ 𝐴𝐵) → 𝐴𝐵)
84, 7impbii 201 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 198  wa 385  wo 874   class class class wbr 4841  cen 8190  cdom 8191  csdm 8192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-xp 5316  df-rel 5317  df-f1o 6106  df-en 8194  df-dom 8195  df-sdom 8196
This theorem is referenced by:  marypha1lem  8579  tskwe  9060  infxpenlem  9120  cdainflem  9299  axcclem  9565  alephsuc3  9688  gchen1  9733  gchen2  9734  inatsk  9886  ufilen  22059  dirith2  25566  f1ocnt  30069  lindsenlbs  33885  mblfinlem1  33927  axccdom  40156  axccd2  40166
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