Metamath Proof Explorer

Theorem opprc2

Description: Expansion of an ordered pair when the second member is a proper class. See also opprc . (Contributed by NM, 15-Nov-1994) (Revised by Mario Carneiro, 26-Apr-2015)

		Ref	Expression
	Assertion	opprc2	$⊢ \neg B \in V \to ⟨A, B⟩ = \emptyset$

Proof

Step	Hyp	Ref	Expression
1		simpr	$⊢ (A \in V \land B \in V) \to B \in V$
2	1	con3i	$⊢ \neg B \in V \to \neg (A \in V \land B \in V)$
3		opprc	$⊢ \neg (A \in V \land B \in V) \to ⟨A, B⟩ = \emptyset$
4	2 3	syl	$⊢ \neg B \in V \to ⟨A, B⟩ = \emptyset$