Theorem mndoisexid 38548

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 4154	. 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ ExId )
2		df-mndo 38546	. 2 ⊢ MndOp = (SemiGrp ∩ ExId )
3	1, 2	eleq2s 2880	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2142   ∩ cin 3903   ExId cexid 38523  SemiGrpcsem 38539  MndOpcmndo 38545

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-in 3911  df-mndo 38546

This theorem is used by:  mndomgmid  38550  rngo1cl  38618