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Metamath Proof Explorer

Theorem fzolb2

Description: The left endpoint of a half-open integer interval is in the set iff the two arguments are integers with M < N . This provides an alternative notation for the "strict upper integer" predicate by analogy to the "weak upper integer" predicate M e. ( ZZ>=N ) . (Contributed by Mario Carneiro, 29-Sep-2015)

		Ref	Expression
	Assertion	fzolb2	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) ↔ 𝑀 < 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		fzolb	⊢ ( 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) ↔ ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) )
2		df-3an	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 < 𝑁 ) ↔ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 < 𝑁 ) )
3	1 2	bitri	⊢ ( 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) ↔ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) ∧ 𝑀 < 𝑁 ) )
4	3	baib	⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ∈ ( 𝑀 ..^ 𝑁 ) ↔ 𝑀 < 𝑁 ) )