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Theorem mndomgmid 21025
Description: A monoid is a magma with an identity element. (Contributed by FL, 18-Feb-2010.) (New usage is discouraged.)
Assertion
Ref Expression
mndomgmid  |-  ( G  e. MndOp  ->  G  e.  (
Magma  i^i  ExId  ) )

Proof of Theorem mndomgmid
StepHypRef Expression
1 mndoismgm 21024 . 2  |-  ( G  e. MndOp  ->  G  e.  Magma )
2 mndoisexid 21023 . 2  |-  ( G  e. MndOp  ->  G  e.  ExId  )
3 elin 3371 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  <-> 
( G  e.  Magma  /\  G  e.  ExId  )
)
41, 2, 3sylanbrc 645 1  |-  ( G  e. MndOp  ->  G  e.  (
Magma  i^i  ExId  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    i^i cin 3164    ExId cexid 20997   Magmacmagm 21001  MndOpcmndo 21020
This theorem is referenced by:  ismndo2  21028  rngoidmlem  21106  expus  25468  clfsebs3  25483  ununr  25523  zintdom  25541  glmrngo  25585  svli2  25587  isdrngo2  26692
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-sgr 21014  df-mndo 21021
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