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Theorem bj-mndsssmgrp 37249
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.
StepHypRef Expression
1 df-mnd 18744 . 2 Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑥𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
21ssrab3 4081 1 Mnd ⊆ Smgrp
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wral 3060  wrex 3069  [wsbc 3787  wss 3950  cfv 6559  (class class class)co 7429  Basecbs 17243  +gcplusg 17293  Smgrpcsgrp 18727  Mndcmnd 18743
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 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-ss 3967  df-mnd 18744
This theorem is referenced by:  bj-mndsssmgrpel  37250
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