Theorem opprc

Description: Expansion of an ordered pair when either member is a proper class. (Contributed by Mario Carneiro, 26-Apr-2015)

	Ref	Expression
Assertion	opprc	$$⊢\neg \left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to ⟨A,B⟩=\varnothing$$

Proof

Step	Hyp	Ref	Expression
1		dfopif	$$⊢⟨A,B⟩=if\left(\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right),\left\{\left\{A\right\},\left\{A,B\right\}\right\},\varnothing \right)$$
2		iffalse	$$⊢\neg \left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to if\left(\left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right),\left\{\left\{A\right\},\left\{A,B\right\}\right\},\varnothing \right)=\varnothing$$
3	1 2	syl5eq	$$⊢\neg \left(A\in \mathrm{V}\wedge B\in \mathrm{V}\right)\to ⟨A,B⟩=\varnothing$$