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Theorem mndoisexid 34000
 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 3951 . 2 (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ ExId )
2 df-mndo 33998 . 2 MndOp = (SemiGrp ∩ ExId )
31, 2eleq2s 2868 1 (𝐺 ∈ MndOp → 𝐺 ∈ ExId )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2145   ∩ cin 3722   ExId cexid 33975  SemiGrpcsem 33991  MndOpcmndo 33997 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-mndo 33998 This theorem is referenced by:  mndomgmid  34002  rngo1cl  34070
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