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Theorem bj-mndsssmgrp 37213
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 18749 . 2 Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑥𝑏 ((𝑒𝑝𝑥) = 𝑥 ∧ (𝑥𝑝𝑒) = 𝑥)}
21ssrab3 4092 1 Mnd ⊆ Smgrp
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1535  wral 3057  wrex 3066  [wsbc 3791  wss 3963  cfv 6558  (class class class)co 7425  Basecbs 17234  +gcplusg 17287  Smgrpcsgrp 18732  Mndcmnd 18748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rab 3433  df-ss 3980  df-mnd 18749
This theorem is referenced by:  bj-mndsssmgrpel  37214
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