Theorem brdom2 9020

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 9016	. . 3 ⊢ ≼ = ( ≺ ∪ ≈ )
2	1	eleq2i 2855	. 2 ⊢ (⟨𝐴, 𝐵⟩ ∈ ≼ ↔ ⟨𝐴, 𝐵⟩ ∈ ( ≺ ∪ ≈ ))
3		df-br 5106	. 2 ⊢ (𝐴 ≼ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≼ )
4		df-br 5106	. . . 4 ⊢ (𝐴 ≺ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≺ )
5		df-br 5106	. . . 4 ⊢ (𝐴 ≈ 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ ≈ )
6	4, 5	orbi12i 928	. . 3 ⊢ ((𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵) ↔ (⟨𝐴, 𝐵⟩ ∈ ≺ ∨ ⟨𝐴, 𝐵⟩ ∈ ≈ ))
7		elun 4154	. . 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 2146   ∪ cun 4151  ⟨cop 4837   class class class wbr 5105   ≈ cen 8981   ≼ cdom 8982   ≺ csdm 8983

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-br 5106  df-opab 5170  df-f1o 6553  df-en 8985  df-dom 8986  df-sdom 8987

This theorem is used by:  bren2  9021  domnsym  9133  domnsymfi  9226  modom  9253  carddom2  10068  axcc4dom  10529  entric  10651  entri2  10652  gchor  10722  frgpcyg  21891  iunmbl2  25890  dyadmbl  25933  padct  33364  volmeas  34917  ovoliunnfl  38720  ctbnfien  43965