Theorem brdom2 8919

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 8915	. . 3 ⊢ ≼ = ( ≺ ∪ ≈ )
2	1	eleq2i 2828	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≼ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
3		df-br 5099	. 2 ⊢ (𝐴 ≼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
4		df-br 5099	. . . 4 ⊢ (𝐴 ≺ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
5		df-br 5099	. . . 4 ⊢ (𝐴 ≈ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≈ )
6	4, 5	orbi12i 914	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
7		elun 4105	. . 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 2113   ∪ cun 3899  ⟨cop 4586   class class class wbr 5098   ≈ cen 8880   ≼ cdom 8881   ≺ csdm 8882

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-br 5099  df-opab 5161  df-f1o 6499  df-en 8884  df-dom 8885  df-sdom 8886

This theorem is referenced by:  bren2  8920  domnsym  9031  domnsymfi  9124  modom  9151  carddom2  9889  axcc4dom  10351  entric  10467  entri2  10468  gchor  10538  frgpcyg  21528  iunmbl2  25514  dyadmbl  25557  padct  32797  volmeas  34388  ovoliunnfl  37863  ctbnfien  43070