Metamath Proof Explorer

Theorem elfzo0l

Description: A member of a half-open range of nonnegative integers is either 0 or a member of the corresponding half-open range of positive integers. (Contributed by AV, 5-Feb-2021)

		Ref	Expression
	Assertion	elfzo0l	$⊢ K \in (0 ..^N) \to (K = 0 \lor K \in (1 ..^N))$

Proof

Step	Hyp	Ref	Expression
1		elfzo0	$⊢ K \in (0 ..^N) \leftrightarrow (K \in ℕ_{0} \land N \in ℕ \land K < N)$
2	1	simp2bi	$⊢ K \in (0 ..^N) \to N \in ℕ$
3		fzo0sn0fzo1	$⊢ N \in ℕ \to (0 ..^N) = \{0\} \cup (1 ..^N)$
4	3	eleq2d	$⊢ N \in ℕ \to (K \in (0 ..^N) \leftrightarrow K \in (\{0\} \cup (1 ..^N)))$
5		elun	$⊢ K \in (\{0\} \cup (1 ..^N)) \leftrightarrow (K \in \{0\} \lor K \in (1 ..^N))$
6		elsni	$⊢ K \in \{0\} \to K = 0$
7	6	orim1i	$⊢ (K \in \{0\} \lor K \in (1 ..^N)) \to (K = 0 \lor K \in (1 ..^N))$
8	5 7	sylbi	$⊢ K \in (\{0\} \cup (1 ..^N)) \to (K = 0 \lor K \in (1 ..^N))$
9	4 8	syl6bi	$⊢ N \in ℕ \to (K \in (0 ..^N) \to (K = 0 \lor K \in (1 ..^N)))$
10	2 9	mpcom	$⊢ K \in (0 ..^N) \to (K = 0 \lor K \in (1 ..^N))$