Theorem bj-mndsssmgrp 37253

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 18668	. 2 ⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑥 ∈ 𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
2	1	ssrab3 4047	1 ⊢ Mnd ⊆ Smgrp

Syntax hints:   ∧ wa 395   = wceq 1540  ∀wral 3045  ∃wrex 3054  [wsbc 3755   ⊆ wss 3916  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  +_gcplusg 17226  Smgrpcsgrp 18651  Mndcmnd 18667

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-ss 3933  df-mnd 18668

This theorem is referenced by:  bj-mndsssmgrpel  37254