Theorem elfzolt3b

Description: Membership in a half-open integer interval implies that the bounds are unequal. (Contributed by Mario Carneiro, 29-Sep-2015)

	Ref	Expression
Assertion	elfzolt3b	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		elfzoel1	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 ∈ ℤ )
2		elfzoel2	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑁 ∈ ℤ )
3		elfzolt3	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 < 𝑁 )
4		fzolb	⊢ ( 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) )
5	1 2 3 4	syl3anbrc	⊢ ( 𝐾 ∈ ( 𝑀 ..^ 𝑁 ) → 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) )