Theorem brdom2 9047

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 9043	. . 3 ⊢ ≼ = ( ≺ ∪ ≈ )
2	1	eleq2i 2856	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≼ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
3		df-br 5109	. 2 ⊢ (𝐴 ≼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
4		df-br 5109	. . . 4 ⊢ (𝐴 ≺ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
5		df-br 5109	. . . 4 ⊢ (𝐴 ≈ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≈ )
6	4, 5	orbi12i 928	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
7		elun 4155	. . 3 ⊢ (⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
8	6, 7	bitr4i 281	. 2 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
9	2, 3, 8	3bitr4i 306	1 ⊢ (𝐴 ≼ 𝐵 ↔ (𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵))

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861   ∈ wcel 2147   ∪ cun 4152  ⟨cop 4840   class class class wbr 5108   ≈ cen 9008   ≼ cdom 9009   ≺ csdm 9010

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-br 5109  df-opab 5174  df-f1o 6573  df-en 9012  df-dom 9013  df-sdom 9014

This theorem is used by:  bren2  9048  domnsym  9160  domnsymfi  9253  modom  9280  carddom2  10095  axcc4dom  10558  entric  10680  entri2  10681  gchor  10751  frgpcyg  21934  iunmbl2  25933  dyadmbl  25976  padct  33407  volmeas  34975  ovoliunnfl  38848  ctbnfien  44091