Metamath Proof Explorer

Theorem sgrpssmgm

Description: The class of all semigroups is a proper subclass of the class of all magmas. (Contributed by AV, 29-Jan-2020)

		Ref	Expression
	Assertion	sgrpssmgm	\|- Smgrp C. Mgm

Proof


 |-  ( x e. Smgrp -> x e. Mgm )
 |-  Smgrp C_ Mgm
 |-  E. x e. Mgm x e/ Smgrp
 |-  ( ( Smgrp C_ Mgm /\ E. x e. Mgm x e/ Smgrp ) -> Smgrp C. Mgm )
 |-  Smgrp C. Mgm

Step	Hyp	Ref	Expression
1		sgrpmgm	\|- ( x e. Smgrp -> x e. Mgm )
2	1	ssriv	\|- Smgrp C_ Mgm
3		mgmnsgrpex	\|- E. x e. Mgm x e/ Smgrp
4		ssexnelpss	\|- ( ( Smgrp C_ Mgm /\ E. x e. Mgm x e/ Smgrp ) -> Smgrp C. Mgm )
5	2 3 4	mp2an	\|- Smgrp C. Mgm