Theorem opthne

Description: Two ordered pairs are not equal iff their first components or their second components are not equal. (Contributed by AV, 13-Dec-2018)

	Ref	Expression
Hypotheses	opthne.1	$$⊢A\in \mathrm{V}$$
	opthne.2	$$⊢B\in \mathrm{V}$$
Assertion	opthne	$$⊢⟨A,B⟩\ne ⟨C,D⟩\leftrightarrow \left(A\ne C\vee B\ne D\right)$$

Proof

Step	Hyp	Ref	Expression
1		opthne.1	$$⊢A\in \mathrm{V}$$
2		opthne.2	$$⊢B\in \mathrm{V}$$
3		opthneg	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to \left(⟨A,B⟩\ne ⟨C,D⟩\leftrightarrow \left(A\ne C\vee B\ne D\right)\right)$$
4	1 2 3	mp2an	$$⊢⟨A,B⟩\ne ⟨C,D⟩\leftrightarrow \left(A\ne C\vee B\ne D\right)$$