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Theorem brsdom2 8069
Description: Alternate definition of strict dominance. Definition 3 of [Suppes] p. 97. (Contributed by NM, 27-Jul-2004.)
Hypotheses
Ref Expression
brsdom2.1 𝐴 ∈ V
brsdom2.2 𝐵 ∈ V
Assertion
Ref Expression
brsdom2 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))

Proof of Theorem brsdom2
StepHypRef Expression
1 dfsdom2 8068 . . 3 ≺ = ( ≼ ∖ ≼ )
21eleq2i 2691 . 2 (⟨𝐴, 𝐵⟩ ∈ ≺ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≼ ))
3 df-br 4645 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
4 df-br 4645 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
5 df-br 4645 . . . . . 6 (𝐵𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ )
6 brsdom2.1 . . . . . . 7 𝐴 ∈ V
7 brsdom2.2 . . . . . . 7 𝐵 ∈ V
86, 7opelcnv 5293 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ ≼ ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ )
95, 8bitr4i 267 . . . . 5 (𝐵𝐴 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
109notbii 310 . . . 4 𝐵𝐴 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ≼ )
114, 10anbi12i 732 . . 3 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ≼ ))
12 eldif 3577 . . 3 (⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≼ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ≼ ))
1311, 12bitr4i 267 . 2 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≼ ))
142, 3, 133bitr4i 292 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wa 384  wcel 1988  Vcvv 3195  cdif 3564  cop 4174   class class class wbr 4644  ccnv 5103  cdom 7938  csdm 7939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943
This theorem is referenced by: (None)
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