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Theorem mndoisexid 32838
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 elin 3758 . . 3 (𝐺 ∈ (SemiGrp ∩ ExId ) ↔ (𝐺 ∈ SemiGrp ∧ 𝐺 ∈ ExId ))
21simprbi 479 . 2 (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ ExId )
3 df-mndo 32836 . 2 MndOp = (SemiGrp ∩ ExId )
42, 3eleq2s 2706 1 (𝐺 ∈ MndOp → 𝐺 ∈ ExId )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1977  cin 3539   ExId cexid 32813  SemiGrpcsem 32829  MndOpcmndo 32835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-mndo 32836
This theorem is referenced by:  mndomgmid  32840  rngo1cl  32908
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