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Theorem bren2 7872
 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 7868 . . 3 (𝐴𝐵𝐴𝐵)
2 sdomnen 7870 . . . 4 (𝐴𝐵 → ¬ 𝐴𝐵)
32con2i 133 . . 3 (𝐴𝐵 → ¬ 𝐴𝐵)
41, 3jca 553 . 2 (𝐴𝐵 → (𝐴𝐵 ∧ ¬ 𝐴𝐵))
5 brdom2 7871 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
65biimpi 205 . . 3 (𝐴𝐵 → (𝐴𝐵𝐴𝐵))
76orcanai 950 . 2 ((𝐴𝐵 ∧ ¬ 𝐴𝐵) → 𝐴𝐵)
84, 7impbii 198 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383   class class class wbr 4583   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-f1o 5811  df-en 7842  df-dom 7843  df-sdom 7844 This theorem is referenced by:  marypha1lem  8222  tskwe  8659  infxpenlem  8719  cdainflem  8896  axcclem  9162  alephsuc3  9281  gchen1  9326  gchen2  9327  inatsk  9479  ufilen  21544  dirith2  25017  f1ocnt  28946  lindsenlbs  32574  mblfinlem1  32616  axccdom  38411  axccd2  38425
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