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Metamath Proof Explorer

Theorem ssex

Description: The subset of a set is also a set. Exercise 3 of TakeutiZaring p. 22. This is one way to express the Axiom of Separation ax-sep (a.k.a. Subset Axiom). (Contributed by NM, 27-Apr-1994)

		Ref	Expression
	Hypothesis	ssex.1	$⊢ B \in V$
	Assertion	ssex	$⊢ A \subseteq B \to A \in V$

Proof

Step	Hyp	Ref	Expression
1		ssex.1	$⊢ B \in V$
2		dfss2	$⊢ A \subseteq B \leftrightarrow A \cap B = A$
3	1	inex2	$⊢ A \cap B \in V$
4		eleq1	$⊢ A \cap B = A \to (A \cap B \in V \leftrightarrow A \in V)$
5	3 4	mpbii	$⊢ A \cap B = A \to A \in V$
6	2 5	sylbi	$⊢ A \subseteq B \to A \in V$