Theorem opab0

Description: Empty ordered pair class abstraction. (Contributed by AV, 29-Oct-2021)

	Ref	Expression
Assertion	opab0	$$⊢\left\{⟨x,y⟩|\phi \right\}=\varnothing \leftrightarrow \forall x\forall y\neg \phi$$

Proof

Step	Hyp	Ref	Expression
1		opabn0	$$⊢\left\{⟨x,y⟩|\phi \right\}\ne \varnothing \leftrightarrow \exists x\exists y\phi$$
2		df-ne	$$⊢\left\{⟨x,y⟩|\phi \right\}\ne \varnothing \leftrightarrow \neg \left\{⟨x,y⟩|\phi \right\}=\varnothing$$
3		2exnaln	$$⊢\exists x\exists y\phi \leftrightarrow \neg \forall x\forall y\neg \phi$$
4	1 2 3	3bitr3i	$$⊢\neg \left\{⟨x,y⟩|\phi \right\}=\varnothing \leftrightarrow \neg \forall x\forall y\neg \phi$$
5	4	con4bii	$$⊢\left\{⟨x,y⟩|\phi \right\}=\varnothing \leftrightarrow \forall x\forall y\neg \phi$$