Theorem bj-cmnssmnd 33136

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

Assertion

Ref	Expression
bj-cmnssmnd	|- CMnd C_ 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 18195	. 2 |- CMnd = { x e. Mnd | A. y e. ( Base ` x ) A. z e. ( Base ` x ) ( y ( +g ` x ) z ) = ( z ( +g ` x ) y ) }
2		ssrab2 3687	. 2 |- { x e. Mnd | A. y e. ( Base ` x ) A. z e. ( Base ` x ) ( y ( +g ` x ) z ) = ( z ( +g ` x ) y ) } C_ Mnd
3	1, 2	eqsstri 3635	1 |- CMnd C_ Mnd

Syntax hints:   = wceq 1483   A.wral 2912   {crab 2916   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941   Mndcmnd 17294  CMndccmn 18193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-in 3581  df-ss 3588  df-cmn 18195

This theorem is referenced by:  bj-cmnssmndel  33137