Theorem mndsssgrp

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

	Ref	Expression
Assertion	mndsssgrp	⊢ Mnd ⊊ Smgrp

Proof

Step	Hyp	Ref	Expression
1		mndsgrp	⊢ ( 𝑥 ∈ Mnd → 𝑥 ∈ Smgrp )
2	1	ssriv	⊢ Mnd ⊆ Smgrp
3		sgrpnmndex	⊢ ∃ 𝑥 ∈ Smgrp 𝑥 ∉ Mnd
4		ssexnelpss	⊢ ( ( Mnd ⊆ Smgrp ∧ ∃ 𝑥 ∈ Smgrp 𝑥 ∉ Mnd ) → Mnd ⊊ Smgrp )
5	2 3 4	mp2an	⊢ Mnd ⊊ Smgrp