Theorem poleloe

Description: Express "less than or equals" for general strict orders. (Contributed by Stefan O'Rear, 17-Jan-2015)

	Ref	Expression
Assertion	poleloe	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ( 𝑅 ∪ I ) 𝐵 ↔ ( 𝐴 𝑅 𝐵 ∨ 𝐴 = 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		brun	⊢ ( 𝐴 ( 𝑅 ∪ I ) 𝐵 ↔ ( 𝐴 𝑅 𝐵 ∨ 𝐴 I 𝐵 ) )
2		ideqg	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 I 𝐵 ↔ 𝐴 = 𝐵 ) )
3	2	orbi2d	⊢ ( 𝐵 ∈ 𝑉 → ( ( 𝐴 𝑅 𝐵 ∨ 𝐴 I 𝐵 ) ↔ ( 𝐴 𝑅 𝐵 ∨ 𝐴 = 𝐵 ) ) )
4	1 3	syl5bb	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 ( 𝑅 ∪ I ) 𝐵 ↔ ( 𝐴 𝑅 𝐵 ∨ 𝐴 = 𝐵 ) ) )