Theorem rabeq0

Description: Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010) (Revised by BJ, 16-Jul-2021)

	Ref	Expression
Assertion	rabeq0	$$⊢\left\{x\in A|\phi \right\}=\varnothing \leftrightarrow \forall x\in A\neg \phi$$

Proof

Step	Hyp	Ref	Expression
1		ab0	$$⊢\left\{x|\left(x\in A\wedge \phi \right)\right\}=\varnothing \leftrightarrow \forall x\neg \left(x\in A\wedge \phi \right)$$
2		df-rab	$$⊢\left\{x\in A|\phi \right\}=\left\{x|\left(x\in A\wedge \phi \right)\right\}$$
3	2	eqeq1i	$$⊢\left\{x\in A|\phi \right\}=\varnothing \leftrightarrow \left\{x|\left(x\in A\wedge \phi \right)\right\}=\varnothing$$
4		raln	$$⊢\forall x\in A\neg \phi \leftrightarrow \forall x\neg \left(x\in A\wedge \phi \right)$$
5	1 3 4	3bitr4i	$$⊢\left\{x\in A|\phi \right\}=\varnothing \leftrightarrow \forall x\in A\neg \phi$$