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

Theorem brsdom2 8291
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 8290 . . 3 ≺ = ( ≼ ∖ ≼ )
21eleq2i 2836 . 2 (⟨𝐴, 𝐵⟩ ∈ ≺ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≼ ))
3 df-br 4810 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
4 df-br 4810 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
5 df-br 4810 . . . . . 6 (𝐵𝐴 ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ )
6 brsdom2.1 . . . . . . 7 𝐴 ∈ V
7 brsdom2.2 . . . . . . 7 𝐵 ∈ V
86, 7opelcnv 5472 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ ≼ ↔ ⟨𝐵, 𝐴⟩ ∈ ≼ )
95, 8bitr4i 269 . . . . 5 (𝐵𝐴 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
109notbii 311 . . . 4 𝐵𝐴 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ≼ )
114, 10anbi12i 620 . . 3 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ≼ ))
12 eldif 3742 . . 3 (⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≼ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ≼ ))
1311, 12bitr4i 269 . 2 ((𝐴𝐵 ∧ ¬ 𝐵𝐴) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≼ ))
142, 3, 133bitr4i 294 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐵𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 197  wa 384  wcel 2155  Vcvv 3350  cdif 3729  cop 4340   class class class wbr 4809  ccnv 5276  cdom 8158  csdm 8159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator