Theorem bj-cmnssmnd 37251

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 19796	. 2 ⊢ CMnd = {𝑥 ∈ Mnd ∣ ∀𝑦 ∈ (Base‘𝑥)∀𝑧 ∈ (Base‘𝑥)(𝑦(+g‘𝑥)𝑧) = (𝑧(+g‘𝑥)𝑦)}
2	1	ssrab3 4081	1 ⊢ CMnd ⊆ Mnd

Syntax hints:   = wceq 1540  ∀wral 3060   ⊆ wss 3950  ‘cfv 6559  (class class class)co 7429  Basecbs 17243  +_gcplusg 17293  Mndcmnd 18743  CMndccmn 19794

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-ss 3967  df-cmn 19796

This theorem is referenced by:  bj-cmnssmndel  37252