Theorem bj-cmnssmnd 37645

Description: Commutative monoids are monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-cmnssmnd	⊢ CMnd ⊆ Mnd

Proof of Theorem bj-cmnssmnd

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cmn 19751	. 2 ⊢ CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g‘𝑥)𝑧) = (𝑧(+g‘𝑥)𝑦)}
2	1	ssrab3 4015	1 ⊢ CMnd ⊆ Mnd

Syntax hints:   = wceq 1548  ∀wral 3055   ⊆ wss 3884  ‘cfv 6488  (class class class)co 7359  Basecbs 17174  +_gcplusg 17215  Mndcmnd 18697  CMndccmn 19749

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-ss 3901  df-cmn 19751

This theorem is referenced by:  bj-cmnssmndel  37646