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Theorem brdom2 9023
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 9019 . . 3 ≼ = ( ≺ ∪ ≈ )
21eleq2i 2832 . 2 (⟨𝐴, 𝐵⟩ ∈ ≼ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
3 df-br 5143 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
4 df-br 5143 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
5 df-br 5143 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≈ )
64, 5orbi12i 914 . . 3 ((𝐴𝐵𝐴𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
7 elun 4152 . . 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 2107  cun 3948  cop 4631   class class class wbr 5142  cen 8983  cdom 8984  csdm 8985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-f1o 6567  df-en 8987  df-dom 8988  df-sdom 8989
This theorem is referenced by:  bren2  9024  domnsym  9140  domnsymfi  9241  modom  9281  carddom2  10018  axcc4dom  10482  entric  10598  entri2  10599  gchor  10668  frgpcyg  21593  iunmbl2  25593  dyadmbl  25636  padct  32732  volmeas  34233  ovoliunnfl  37670  ctbnfien  42834
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