Theorem opprc1

Description: Expansion of an ordered pair when the first member is a proper class. See also opprc . (Contributed by NM, 10-Apr-2004) (Revised by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Assertion	opprc1	$$⊢\neg A\in \mathrm{V}\to ⟨A,B⟩=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		simpl	$$⊢\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to A\in \mathrm{V}$$
2	1	con3i	$$⊢\neg A\in \mathrm{V}\to \neg \left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)$$
3		opprc	$$⊢\neg \left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to ⟨A,B⟩=\varnothing$$
4	2 3	syl	$$⊢\neg A\in \mathrm{V}\to ⟨A,B⟩=\varnothing$$