MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  brsdom2 Structured version   Visualization version   GIF version

Theorem brsdom2 9137
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 9136 . . 3 ≺ = ( ≼ ∖ ≼ )
21eleq2i 2833 . 2 (⟨𝐴, 𝐵⟩ ∈ ≺ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≼ ))
3 df-br 5144 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
4 df-br 5144 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
5 df-br 5144 . . . . . 6 (𝐵𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ )
6 brsdom2.1 . . . . . . 7 𝐴 ∈ V
7 brsdom2.2 . . . . . . 7 𝐵 ∈ V
86, 7opelcnv 5892 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ ≼ ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ )
95, 8bitr4i 278 . . . . 5 (𝐵𝐴 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
109notbii 320 . . . 4 𝐵𝐴 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ≼ )
114, 10anbi12i 628 . . 3 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ≼ ))
12 eldif 3961 . . 3 (⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≼ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ≼ ))
1311, 12bitr4i 278 . 2 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≼ ))
142, 3, 133bitr4i 303 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395  wcel 2108  Vcvv 3480  cdif 3948  cop 4632   class class class wbr 5143  ccnv 5684  cdom 8983  csdm 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator