Theorem bj-grpssmnd 37257

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 18874	. 2 ⊢ Grp = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∃𝑧 ∈ (Base‘𝑥)(𝑧(+g‘𝑥)𝑦) = (0g‘𝑥)}
2	1	ssrab3 4047	1 ⊢ Grp ⊆ Mnd

Syntax hints:   = wceq 1540  ∀wral 3045  ∃wrex 3054   ⊆ wss 3916  ‘cfv 6513  (class class class)co 7389  Basecbs 17185  +_gcplusg 17226  0_gc0g 17408  Mndcmnd 18667  Grpcgrp 18871

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-ss 3933  df-grp 18874

This theorem is referenced by:  bj-grpssmndel  37258