Theorem bj-mndsssmgrpel 34577

Description: Monoids are semigroups (elemental version). (Contributed by BJ, 11-Apr-2024.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-mndsssmgrpel	⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)

Proof of Theorem bj-mndsssmgrpel

Step	Hyp	Ref	Expression
1		bj-mndsssmgrp 34576	. 2 ⊢ Mnd ⊆ Smgrp
2	1	sseli 3956	1 ⊢ (𝐺 ∈ Mnd → 𝐺 ∈ Smgrp)

Syntax hints:   → wi 4   ∈ wcel 2113  Smgrpcsgrp 17895  Mndcmnd 17906

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rab 3146  df-in 3936  df-ss 3945  df-mnd 17907

This theorem is referenced by: (None)