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	$$⊢B\in \mathrm{V}$$
Assertion	ssex	$$⊢A\subseteq B\to A\in \mathrm{V}$$

Proof

Step	Hyp	Ref	Expression
1		ssex.1	$$⊢B\in \mathrm{V}$$
2		df-ss	$$⊢A\subseteq B\leftrightarrow A\cap B=A$$
3	1	inex2	$$⊢A\cap B\in \mathrm{V}$$
4		eleq1	$$⊢A\cap B=A\to \left(A\cap B\in \mathrm{V}\leftrightarrow A\in \mathrm{V}\right)$$
5	3 4	mpbii	$$⊢A\cap B=A\to A\in \mathrm{V}$$
6	2 5	sylbi	$$⊢A\subseteq B\to A\in \mathrm{V}$$