Theorem bj-mndsssmgrp 38285

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

Assertion

Ref	Expression
bj-mndsssmgrp	⊢ Mnd ⊆ Smgrp

Proof of Theorem bj-mndsssmgrp

Dummy variables 𝑔 𝑏 𝑝 𝑒 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-mnd 18930	. 2 ⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
2	1	ssrab3 4007	1 ⊢ Mnd ⊆ Smgrp

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1569  ∀wral 3076  ∃wrex 3086  [wsbc 3738   ⊆ wss 3895  ‘cfv 6539  (class class class)co 7421  Basecbs 17393  +_gcplusg 17434  Smgrpcsgrp 18913  Mndcmnd 18929

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-ss 3896  df-mnd 18930

This theorem is used by:  bj-mndsssmgrpel  38286