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Theorem mndoissmgrp 21958
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 3516 . . 3  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  <-> 
( G  e.  SemiGrp  /\  G  e.  ExId  ) )
21simplbi 448 . 2  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  ->  G  e.  SemiGrp )
3 df-mndo 21957 . 2  |- MndOp  =  (
SemiGrp  i^i  ExId  )
42, 3eleq2s 2534 1  |-  ( G  e. MndOp  ->  G  e.  SemiGrp )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1727    i^i cin 3305    ExId cexid 21933   SemiGrpcsem 21949  MndOpcmndo 21956
This theorem is referenced by:  mndoismgm  21960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-v 2964  df-in 3313  df-mndo 21957
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