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Theorem mndoisexid 21023
Description: A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Assertion
Ref Expression
mndoisexid  |-  ( G  e. MndOp  ->  G  e.  ExId  )

Proof of Theorem mndoisexid
StepHypRef Expression
1 elin 3371 . . 3  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  <-> 
( G  e.  SemiGrp  /\  G  e.  ExId  ) )
21simprbi 450 . 2  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  ->  G  e.  ExId  )
3 df-mndo 21021 . 2  |- MndOp  =  (
SemiGrp  i^i  ExId  )
42, 3eleq2s 2388 1  |-  ( G  e. MndOp  ->  G  e.  ExId  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    i^i cin 3164    ExId cexid 20997   SemiGrpcsem 21013  MndOpcmndo 21020
This theorem is referenced by:  mndomgmid  21025  rngo1cl  21112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172  df-mndo 21021
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