Theorem bj-grpssmnd 37641

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 18910	. 2 ⊢ Grp = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∃𝑧 ∈ (Base‘𝑥)(𝑧(+g‘𝑥)𝑦) = (0g‘𝑥)}
2	1	ssrab3 4020	1 ⊢ Grp ⊆ Mnd

Syntax hints:   = wceq 1547  ∀wral 3054  ∃wrex 3064   ⊆ wss 3890  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  0_gc0g 17400  Mndcmnd 18700  Grpcgrp 18907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-rab 3393  df-ss 3907  df-grp 18910

This theorem is referenced by:  bj-grpssmndel  37642