Theorem bj-cmnssmnd 37230

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 19818	. 2 ⊢ CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g‘𝑥)𝑧) = (𝑧(+g‘𝑥)𝑦)}
2	1	ssrab3 4105	1 ⊢ CMnd ⊆ Mnd

Syntax hints:   = wceq 1537  ∀wral 3067   ⊆ wss 3976  ‘cfv 6568  (class class class)co 7443  Basecbs 17252  +_gcplusg 17305  Mndcmnd 18766  CMndccmn 19816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-ss 3993  df-cmn 19818

This theorem is referenced by:  bj-cmnssmndel  37231