Theorem brdom2

Description: Dominance in terms of strict dominance and equinumerosity. Theorem 22(iv) of Suppes p. 97. (Contributed by NM, 17-Jun-1998)

	Ref	Expression
Assertion	brdom2	⊢ ( 𝐴 ≼ 𝐵 ↔ ( 𝐴 ≺ 𝐵 ∨ 𝐴 ≈ 𝐵 ) )

Proof

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