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Theorem brdom2 8102
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 8098 . . 3 ≼ = ( ≺ ∪ ≈ )
21eleq2i 2795 . 2 (⟨𝐴, 𝐵⟩ ∈ ≼ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
3 df-br 4761 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
4 df-br 4761 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
5 df-br 4761 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≈ )
64, 5orbi12i 544 . . 3 ((𝐴𝐵𝐴𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
7 elun 3861 . . 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 382  wcel 2103  cun 3678  cop 4291   class class class wbr 4760  cen 8069  cdom 8070  csdm 8071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ral 3019  df-rab 3023  df-v 3306  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-br 4761  df-opab 4821  df-xp 5224  df-rel 5225  df-f1o 6008  df-en 8073  df-dom 8074  df-sdom 8075
This theorem is referenced by:  bren2  8103  domnsym  8202  modom  8277  carddom2  8916  axcc4dom  9376  entric  9492  entri2  9493  gchor  9562  frgpcyg  20045  iunmbl2  23446  dyadmbl  23489  padct  29727  volmeas  30524  ovoliunnfl  33683  ctbnfien  37801
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