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Theorem bj-grpssmnd 37253
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 18950 . 2 Grp = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∃𝑧 ∈ (Base‘𝑥)(𝑧(+g𝑥)𝑦) = (0g𝑥)}
21ssrab3 4081 1 Grp ⊆ Mnd
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wral 3060  wrex 3069  wss 3950  cfv 6559  (class class class)co 7429  Basecbs 17243  +gcplusg 17293  0gc0g 17480  Mndcmnd 18743  Grpcgrp 18947
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-grp 18950
This theorem is referenced by:  bj-grpssmndel  37254
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