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Theorem bj-mndsssmgrp 36151
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 18626 . 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 397   = wceq 1542  wral 3062  wrex 3071  [wsbc 3778  wss 3949  cfv 6544  (class class class)co 7409  Basecbs 17144  +gcplusg 17197  Smgrpcsgrp 18609  Mndcmnd 18625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956  df-ss 3966  df-mnd 18626
This theorem is referenced by:  bj-mndsssmgrpel  36152
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