Theorem bj-mndsssmgrp 37631

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 18701	. 2 ⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
2	1	ssrab3 4020	1 ⊢ Mnd ⊆ Smgrp

Syntax hints:   ∧ wa 396   = wceq 1547  ∀wral 3054  ∃wrex 3064  [wsbc 3730   ⊆ wss 3890  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  Smgrpcsgrp 18684  Mndcmnd 18700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-ss 3907  df-mnd 18701

This theorem is referenced by:  bj-mndsssmgrpel  37632