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Theorem mndoisexid 36378
Description: A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)
Assertion
Ref Expression
mndoisexid (𝐺 ∈ MndOp → 𝐺 ∈ ExId )

Proof of Theorem mndoisexid
StepHypRef Expression
1 elinel2 4160 . 2 (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ ExId )
2 df-mndo 36376 . 2 MndOp = (SemiGrp ∩ ExId )
31, 2eleq2s 2852 1 (𝐺 ∈ MndOp → 𝐺 ∈ ExId )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  cin 3913   ExId cexid 36353  SemiGrpcsem 36369  MndOpcmndo 36375
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-in 3921  df-mndo 36376
This theorem is referenced by:  mndomgmid  36380  rngo1cl  36448
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