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Theorem mndoisexid 21920
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 3522 . . 3  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  <-> 
( G  e.  SemiGrp  /\  G  e.  ExId  ) )
21simprbi 451 . 2  |-  ( G  e.  ( SemiGrp  i^i  ExId  )  ->  G  e.  ExId  )
3 df-mndo 21918 . 2  |- MndOp  =  (
SemiGrp  i^i  ExId  )
42, 3eleq2s 2527 1  |-  ( G  e. MndOp  ->  G  e.  ExId  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    i^i cin 3311    ExId cexid 21894   SemiGrpcsem 21910  MndOpcmndo 21917
This theorem is referenced by:  mndomgmid  21922  rngo1cl  22009
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-in 3319  df-mndo 21918
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