Theorem eq0f

Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of Suppes p. 22. (Contributed by BJ, 15-Jul-2021)

	Ref	Expression
Hypothesis	eq0f.1	$$⊢\underset{\_}{Ⅎ}xA$$
Assertion	eq0f	$$⊢A=\varnothing \leftrightarrow \forall x\neg x\in A$$

Proof

Step	Hyp	Ref	Expression
1		eq0f.1	$$⊢\underset{\_}{Ⅎ}xA$$
2		nfcv	$$⊢\underset{\_}{Ⅎ}x\varnothing$$
3	1 2	cleqf	$$⊢A=\varnothing \leftrightarrow \forall x\left(x\in A\leftrightarrow x\in \varnothing \right)$$
4		noel	$$⊢\neg x\in \varnothing$$
5	4	nbn	$$⊢\neg x\in A\leftrightarrow \left(x\in A\leftrightarrow x\in \varnothing \right)$$
6	5	albii	$$⊢\forall x\neg x\in A\leftrightarrow \forall x\left(x\in A\leftrightarrow x\in \varnothing \right)$$
7	3 6	bitr4i	$$⊢A=\varnothing \leftrightarrow \forall x\neg x\in A$$