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Theorem bj-mndsssmgrp 36455
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 18661 . 2 Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑥𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
21ssrab3 4080 1 Mnd ⊆ Smgrp
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wral 3060  wrex 3069  [wsbc 3777  wss 3948  cfv 6543  (class class class)co 7412  Basecbs 17149  +gcplusg 17202  Smgrpcsgrp 18644  Mndcmnd 18660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-in 3955  df-ss 3965  df-mnd 18661
This theorem is referenced by:  bj-mndsssmgrpel  36456
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