Theorem nelpr1

Description: If a class is not an element of an unordered pair, it is not the first listed element. (Contributed by Glauco Siliprandi, 3-Mar-2021)

	Ref	Expression
Hypotheses	nelpr1.a	$$⊢\phi \to A\in V$$
	nelpr1.n	$$⊢\phi \to \neg A\in \left\{B,C\right\}$$
Assertion	nelpr1	$$⊢\phi \to A\ne B$$

Proof

Step	Hyp	Ref	Expression
1		nelpr1.a	$$⊢\phi \to A\in V$$
2		nelpr1.n	$$⊢\phi \to \neg A\in \left\{B,C\right\}$$
3		animorrl	$$⊢\left(\phi \wedge A=B\right)\to \left(A=B\vee A=C\right)$$
4		elprg	$$⊢A\in V\to \left(A\in \left\{B,C\right\}\leftrightarrow \left(A=B\vee A=C\right)\right)$$
5	1 4	syl	$$⊢\phi \to \left(A\in \left\{B,C\right\}\leftrightarrow \left(A=B\vee A=C\right)\right)$$
6	5	adantr	$$⊢\left(\phi \wedge A=B\right)\to \left(A\in \left\{B,C\right\}\leftrightarrow \left(A=B\vee A=C\right)\right)$$
7	3 6	mpbird	$$⊢\left(\phi \wedge A=B\right)\to A\in \left\{B,C\right\}$$
8	2 7	mtand	$$⊢\phi \to \neg A=B$$
9	8	neqned	$$⊢\phi \to A\ne B$$