eq0

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 NM, 29-Aug-1993) Avoid ax-11 , ax-12 . (Revised by GG and Steven Nguyen, 28-Jun-2024) Avoid ax-8 , df-clel . (Revised by GG, 6-Sep-2024)

		Ref	Expression
	Assertion	eq0	$⊢ A = \emptyset \leftrightarrow \forall x \neg x \in A$

Proof

Step	Hyp	Ref	Expression
1		biidd	$⊢ y = x \to (⊥ \leftrightarrow ⊥)$
2	1	eqabbw	$⊢ A = \{y \| ⊥\} \leftrightarrow \forall x (x \in A \leftrightarrow ⊥)$
3		dfnul4	$⊢ \emptyset = \{y \| ⊥\}$
4	3	eqeq2i	$⊢ A = \emptyset \leftrightarrow A = \{y \| ⊥\}$
5		nbfal	$⊢ \neg x \in A \leftrightarrow (x \in A \leftrightarrow ⊥)$
6	5	albii	$⊢ \forall x \neg x \in A \leftrightarrow \forall x (x \in A \leftrightarrow ⊥)$
7	2 4 6	3bitr4i	$⊢ A = \emptyset \leftrightarrow \forall x \neg x \in A$

Metamath Proof Explorer

Theorem eq0

Proof