Theorem bj-grpssmnd 37233

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.

Step	Hyp	Ref	Expression
1		df-grp 18970	. 2 ⊢ Grp = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∃𝑧 ∈ (Base‘𝑥)(𝑧(+g‘𝑥)𝑦) = (0g‘𝑥)}
2	1	ssrab3 4105	1 ⊢ Grp ⊆ Mnd

Syntax hints:   = wceq 1537  ∀wral 3067  ∃wrex 3076   ⊆ wss 3976  ‘cfv 6568  (class class class)co 7443  Basecbs 17252  +_gcplusg 17305  0_gc0g 17493  Mndcmnd 18766  Grpcgrp 18967

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-ss 3993  df-grp 18970

This theorem is referenced by:  bj-grpssmndel  37234