Mathbox for BJ < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ablsscmn Structured version   Visualization version   GIF version

Theorem bj-ablsscmn 32770
 Description: Abelian groups are commutative monoids. (Contributed by BJ, 9-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ablsscmn Abel ⊆ CMnd

Proof of Theorem bj-ablsscmn
StepHypRef Expression
1 df-abl 18117 . 2 Abel = (Grp ∩ CMnd)
2 inss2 3812 . 2 (Grp ∩ CMnd) ⊆ CMnd
31, 2eqsstri 3614 1 Abel ⊆ CMnd
 Colors of variables: wff setvar class Syntax hints:   ∩ cin 3554   ⊆ wss 3555  Grpcgrp 17343  CMndccmn 18114  Abelcabl 18115 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-in 3562  df-ss 3569  df-abl 18117 This theorem is referenced by:  bj-ablsscmnel  32771  bj-rrvecsscmn  32782
 Copyright terms: Public domain W3C validator