MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mndoissmgrp Unicode version

Theorem mndoissmgrp 21006
Description: A monoid is a semi-group. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Assertion
Ref Expression
mndoissmgrp  |-  ( G  e. MndOp  ->  G  e.  SemiGrp )

Proof of Theorem mndoissmgrp
StepHypRef Expression
1 elin 3358 . . 3  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  <-> 
( G  e.  SemiGrp  /\  G  e.  ExId  ) )
21simplbi 446 . 2  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  ->  G  e.  SemiGrp )
3 df-mndo 21005 . 2  |- MndOp  =  (
SemiGrp  i^i  ExId  )
42, 3eleq2s 2375 1  |-  ( G  e. MndOp  ->  G  e.  SemiGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    i^i cin 3151    ExId cexid 20981   SemiGrpcsem 20997  MndOpcmndo 21004
This theorem is referenced by:  mndoismgm  21008  mndoisass  25356
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-mndo 21005
  Copyright terms: Public domain W3C validator