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Theorem brdom2 7871
Description: Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of [Suppes] p. 97. (Contributed by NM, 17-Jun-1998.)
Assertion
Ref Expression
brdom2 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))

Proof of Theorem brdom2
StepHypRef Expression
1 dfdom2 7867 . . 3 ≼ = ( ≺ ∪ ≈ )
21eleq2i 2680 . 2 (⟨𝐴, 𝐵⟩ ∈ ≼ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
3 df-br 4584 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
4 df-br 4584 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
5 df-br 4584 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≈ )
64, 5orbi12i 542 . . 3 ((𝐴𝐵𝐴𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
7 elun 3715 . . 3 (⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
86, 7bitr4i 266 . 2 ((𝐴𝐵𝐴𝐵) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
92, 3, 83bitr4i 291 1 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wo 382  wcel 1977  cun 3538  cop 4131   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:  bren2  7872  domnsym  7971  modom  8046  carddom2  8686  axcc4dom  9146  entric  9258  entri2  9259  gchor  9328  frgpcyg  19741  iunmbl2  23132  dyadmbl  23174  padct  28885  volmeas  29621  ovoliunnfl  32621  ctbnfien  36400
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