ssexnelpss

Description: If there is an element of a class which is not contained in a subclass, the subclass is a proper subclass. (Contributed by AV, 29-Jan-2020)

		Ref	Expression
	Assertion	ssexnelpss	$⊢ (A \subseteq B \land \exists x \in B x \notin A) \to A \subset B$

Proof

Step	Hyp	Ref	Expression
1		df-nel	$⊢ x \notin A \leftrightarrow \neg x \in A$
2		ssnelpss	$⊢ A \subseteq B \to ((x \in B \land \neg x \in A) \to A \subset B)$
3	2	expdimp	$⊢ (A \subseteq B \land x \in B) \to (\neg x \in A \to A \subset B)$
4	1 3	biimtrid	$⊢ (A \subseteq B \land x \in B) \to (x \notin A \to A \subset B)$
5	4	rexlimdva	$⊢ A \subseteq B \to (\exists x \in B x \notin A \to A \subset B)$
6	5	imp	$⊢ (A \subseteq B \land \exists x \in B x \notin A) \to A \subset B$

Metamath Proof Explorer

Theorem ssexnelpss

Proof