Theorem lejoin2

Description: A join's second argument is less than or equal to the join. (Contributed by NM, 16-Sep-2011)

	Ref	Expression
Hypotheses	joinval2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	joinval2.l	⊢ ≤ = ( le ‘ 𝐾 )
	joinval2.j	⊢ ∨ = ( join ‘ 𝐾 )
	joinval2.k	⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
	joinval2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
	joinval2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
	joinlem.e	⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
Assertion	lejoin2	⊢ ( 𝜑 → 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		joinval2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		joinval2.l	⊢ ≤ = ( le ‘ 𝐾 )
3		joinval2.j	⊢ ∨ = ( join ‘ 𝐾 )
4		joinval2.k	⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
5		joinval2.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6		joinval2.y	⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7		joinlem.e	⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
8	1 2 3 4 5 6 7	joinlem	⊢ ( 𝜑 → ( ( 𝑋 ≤ ( 𝑋 ∨ 𝑌 ) ∧ 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → ( 𝑋 ∨ 𝑌 ) ≤ 𝑧 ) ) )
9	8	simplrd	⊢ ( 𝜑 → 𝑌 ≤ ( 𝑋 ∨ 𝑌 ) )