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	$$⊢\neg B\in \mathrm{V}\to ⟨A,B⟩=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		simpr	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to B\in \mathrm{V}$$
2		opprc	$$⊢\neg \left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to ⟨A,B⟩=\varnothing$$
3	1 2	nsyl5	$$⊢\neg B\in \mathrm{V}\to ⟨A,B⟩=\varnothing$$