Theorem bj-grpssmnd 36809

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 18895	. 2 ⊢ Grp = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∃𝑧 ∈ (Base‘𝑥)(𝑧(+g‘𝑥)𝑦) = (0g‘𝑥)}
2	1	ssrab3 4072	1 ⊢ Grp ⊆ Mnd

Syntax hints:   = wceq 1533  ∀wral 3051  ∃wrex 3060   ⊆ wss 3940  ‘cfv 6542  (class class class)co 7415  Basecbs 17177  +_gcplusg 17230  0_gc0g 17418  Mndcmnd 18691  Grpcgrp 18892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-ss 3957  df-grp 18895

This theorem is referenced by:  bj-grpssmndel  36810