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Theorem mndomgmid 21932
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 21931 . 2  |-  ( G  e. MndOp  ->  G  e.  Magma )
2 mndoisexid 21930 . 2  |-  ( G  e. MndOp  ->  G  e.  ExId  )
3 elin 3532 . 2  |-  ( G  e.  ( Magma  i^i  ExId  )  <-> 
( G  e.  Magma  /\  G  e.  ExId  )
)
41, 2, 3sylanbrc 647 1  |-  ( G  e. MndOp  ->  G  e.  (
Magma  i^i  ExId  ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1726    i^i cin 3321    ExId cexid 21904   Magmacmagm 21908  MndOpcmndo 21927
This theorem is referenced by:  ismndo2  21935  rngoidmlem  22013  isdrngo2  26576
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
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 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-in 3329  df-sgr 21921  df-mndo 21928
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