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Theorem bj-grpssmnd 36676
Description: Groups are monoids. (Contributed by BJ, 5-Jan-2024.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-grpssmnd Grp ⊆ Mnd

Proof of Theorem bj-grpssmnd
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-grp 18878 . 2 Grp = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∃𝑧 ∈ (Base‘𝑥)(𝑧(+g𝑥)𝑦) = (0g𝑥)}
21ssrab3 4076 1 Grp ⊆ Mnd
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wral 3056  wrex 3065  wss 3944  cfv 6542  (class class class)co 7414  Basecbs 17165  +gcplusg 17218  0gc0g 17406  Mndcmnd 18679  Grpcgrp 18875
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 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rab 3428  df-v 3471  df-in 3951  df-ss 3961  df-grp 18878
This theorem is referenced by:  bj-grpssmndel  36677
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