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Theorem brsdom 8971
Description: Strict dominance relation, meaning "𝐵 is strictly greater in size than 𝐴". Definition of [Mendelson] p. 255. (Contributed by NM, 25-Jun-1998.)
Assertion
Ref Expression
brsdom (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))

Proof of Theorem brsdom
StepHypRef Expression
1 df-sdom 8942 . . 3 ≺ = ( ≼ ∖ ≈ )
21eleq2i 2826 . 2 (⟨𝐴, 𝐵⟩ ∈ ≺ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≈ ))
3 df-br 5150 . 2 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
4 df-br 5150 . . . 4 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
5 df-br 5150 . . . . 5 (𝐴𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≈ )
65notbii 320 . . . 4 𝐴𝐵 ↔ ¬ ⟨𝐴, 𝐵⟩ ∈ ≈ )
74, 6anbi12i 628 . . 3 ((𝐴𝐵 ∧ ¬ 𝐴𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
8 eldif 3959 . . 3 (⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≈ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≼ ∧ ¬ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
97, 8bitr4i 278 . 2 ((𝐴𝐵 ∧ ¬ 𝐴𝐵) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≼ ∖ ≈ ))
102, 3, 93bitr4i 303 1 (𝐴𝐵 ↔ (𝐴𝐵 ∧ ¬ 𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 397  wcel 2107  cdif 3946  cop 4635   class class class wbr 5149  cen 8936  cdom 8937  csdm 8938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-br 5150  df-sdom 8942
This theorem is referenced by:  sdomdom  8976  sdomnen  8977  0sdomg  9104  0sdomgOLD  9105  sdom0  9108  sdomdomtr  9110  domsdomtr  9112  domtriord  9123  canth2  9130  sdomdomtrfi  9204  domsdomtrfi  9205  php2  9211  php2OLD  9223  php3OLD  9224  nnsdomo  9234  1sdom2  9240  sdom1  9242  1sdom2dom  9247  nnsdomg  9302  nnsdomgOLD  9303  card2inf  9550  cardsdomelir  9968  cardsdom2  9983  fidomtri2  9989  cardmin2  9994  alephordi  10069  alephord  10070  isfin4p1  10310  isfin5-2  10386  canthnum  10644  canthwe  10646  canthp1  10649  gchdjuidm  10663  gchxpidm  10664  gchhar  10674  axgroth6  10823  hashsdom  14341  ruc  16186  iscard5  42287
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