Theorem mndoisexid 37919

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

Step	Hyp	Ref	Expression
1		elinel2 4149	. 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ ExId )
2		df-mndo 37917	. 2 ⊢ MndOp = (SemiGrp ∩ ExId )
3	1, 2	eleq2s 2849	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId )

Syntax hints:   → wi 4   ∈ wcel 2111   ∩ cin 3896   ExId cexid 37894  SemiGrpcsem 37910  MndOpcmndo 37916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-in 3904  df-mndo 37917

This theorem is referenced by:  mndomgmid  37921  rngo1cl  37989