Theorem brdom2 8907

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	⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵))

Proof of Theorem brdom2

Step	Hyp	Ref	Expression
1		dfdom2 8903	. . 3 ⊢ ≼ = ( ≺ ∪ ≈ )
2	1	eleq2i 2820	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≼ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
3		df-br 5093	. 2 ⊢ (𝐴 ≼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
4		df-br 5093	. . . 4 ⊢ (𝐴 ≺ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
5		df-br 5093	. . . 4 ⊢ (𝐴 ≈ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≈ )
6	4, 5	orbi12i 914	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
7		elun 4104	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
8	6, 7	bitr4i 278	. 2 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
9	2, 3, 8	3bitr4i 303	1 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵))

Syntax hints:   ↔ wb 206   ∨ wo 847   ∈ wcel 2109   ∪ cun 3901  ⟨cop 4583   class class class wbr 5092   ≈ cen 8869   ≼ cdom 8870   ≺ csdm 8871

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5093  df-opab 5155  df-xp 5625  df-rel 5626  df-f1o 6489  df-en 8873  df-dom 8874  df-sdom 8875

This theorem is referenced by:  bren2  8908  domnsym  9020  domnsymfi  9114  modom  9140  carddom2  9873  axcc4dom  10335  entric  10451  entri2  10452  gchor  10521  frgpcyg  21480  iunmbl2  25456  dyadmbl  25499  padct  32662  volmeas  34198  ovoliunnfl  37646  ctbnfien  42795