Theorem bj-grpssmnd 37250

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 18924	. 2 ⊢ Grp = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∃𝑧 ∈ (Base‘𝑥)(𝑧(+g‘𝑥)𝑦) = (0g‘𝑥)}
2	1	ssrab3 4062	1 ⊢ Grp ⊆ Mnd

Syntax hints:   = wceq 1539  ∀wral 3050  ∃wrex 3059   ⊆ wss 3931  ‘cfv 6541  (class class class)co 7413  Basecbs 17230  +_gcplusg 17274  0_gc0g 17456  Mndcmnd 18717  Grpcgrp 18921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-ss 3948  df-grp 18924

This theorem is referenced by:  bj-grpssmndel  37251