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Theorem brdom2 9021
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 9017 . . 3 ≼ = ( ≺ ∪ ≈ )
21eleq2i 2831 . 2 (⟨𝐴, 𝐵⟩ ∈ ≼ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
3 df-br 5149 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
4 df-br 5149 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
5 df-br 5149 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≈ )
64, 5orbi12i 914 . . 3 ((𝐴𝐵𝐴𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
7 elun 4163 . . 3 (⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
86, 7bitr4i 278 . 2 ((𝐴𝐵𝐴𝐵) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
92, 3, 83bitr4i 303 1 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wo 847  wcel 2106  cun 3961  cop 4637   class class class wbr 5148  cen 8981  cdom 8982  csdm 8983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-f1o 6570  df-en 8985  df-dom 8986  df-sdom 8987
This theorem is referenced by:  bren2  9022  domnsym  9138  domnsymfi  9238  modom  9278  carddom2  10015  axcc4dom  10479  entric  10595  entri2  10596  gchor  10665  frgpcyg  21610  iunmbl2  25606  dyadmbl  25649  padct  32737  volmeas  34212  ovoliunnfl  37649  ctbnfien  42806
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