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Theorem bren2 8152
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 8148 . . 3 (𝐴𝐵𝐴𝐵)
2 sdomnen 8150 . . . 4 (𝐴𝐵 → ¬ 𝐴𝐵)
32con2i 134 . . 3 (𝐴𝐵 → ¬ 𝐴𝐵)
41, 3jca 555 . 2 (𝐴𝐵 → (𝐴𝐵 ∧ ¬ 𝐴𝐵))
5 brdom2 8151 . . . 4 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
65biimpi 206 . . 3 (𝐴𝐵 → (𝐴𝐵𝐴𝐵))
76orcanai 990 . 2 ((𝐴𝐵 ∧ ¬ 𝐴𝐵) → 𝐴𝐵)
84, 7impbii 199 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 382  wa 383   class class class wbr 4804  cen 8118  cdom 8119  csdm 8120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-f1o 6056  df-en 8122  df-dom 8123  df-sdom 8124
This theorem is referenced by:  marypha1lem  8504  tskwe  8966  infxpenlem  9026  cdainflem  9205  axcclem  9471  alephsuc3  9594  gchen1  9639  gchen2  9640  inatsk  9792  ufilen  21935  dirith2  25416  f1ocnt  29868  lindsenlbs  33717  mblfinlem1  33759  axccdom  39915  axccd2  39929
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