Theorem subgss

Description: A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014)

	Ref	Expression
Hypothesis	issubg.b	$$⊢B={\mathrm{Base}}_G$$
Assertion	subgss	$$⊢S\in \mathrm{SubGrp}\left(G\right)\to S\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		issubg.b	$$⊢B={\mathrm{Base}}_G$$
2	1	issubg	$$⊢S\in \mathrm{SubGrp}\left(G\right)\leftrightarrow \left(G\in \mathrm{Grp}\wedge S\subseteq B\wedge G{\upharpoonright }_{𝑠}S\in \mathrm{Grp}\right)$$
3	2	simp2bi	$$⊢S\in \mathrm{SubGrp}\left(G\right)\to S\subseteq B$$