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Theorem brdom2 7945
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 7941 . . 3 ≼ = ( ≺ ∪ ≈ )
21eleq2i 2690 . 2 (⟨𝐴, 𝐵⟩ ∈ ≼ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
3 df-br 4624 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
4 df-br 4624 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
5 df-br 4624 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≈ )
64, 5orbi12i 543 . . 3 ((𝐴𝐵𝐴𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
7 elun 3737 . . 3 (⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
86, 7bitr4i 267 . 2 ((𝐴𝐵𝐴𝐵) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
92, 3, 83bitr4i 292 1 (𝐴𝐵 ↔ (𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wo 383  wcel 1987  cun 3558  cop 4161   class class class wbr 4623  cen 7912  cdom 7913  csdm 7914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-f1o 5864  df-en 7916  df-dom 7917  df-sdom 7918
This theorem is referenced by:  bren2  7946  domnsym  8046  modom  8121  carddom2  8763  axcc4dom  9223  entric  9339  entri2  9340  gchor  9409  frgpcyg  19862  iunmbl2  23265  dyadmbl  23308  padct  29381  volmeas  30117  ovoliunnfl  33122  ctbnfien  36901
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