Theorem bj-cmnssmnd 37860

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 19851	. 2 ⊢ CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g‘𝑥)𝑧) = (𝑧(+g‘𝑥)𝑦)}
2	1	ssrab3 4035	1 ⊢ CMnd ⊆ Mnd

Syntax hints:   = wceq 1568  ∀wral 3077   ⊆ wss 3904  ‘cfv 6536  (class class class)co 7410  Basecbs 17268  +_gcplusg 17309  Mndcmnd 18791  CMndccmn 19849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-ss 3921  df-cmn 19851

This theorem is referenced by:  bj-cmnssmndel  37861