Theorem mndoisexid 38236

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 4131	. 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ ExId )
2		df-mndo 38234	. 2 ⊢ MndOp = (SemiGrp ∩ ExId )
3	1, 2	eleq2s 2857	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId )

Syntax hints:   → wi 4   ∈ wcel 2119   ∩ cin 3882   ExId cexid 38211  SemiGrpcsem 38227  MndOpcmndo 38233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-in 3890  df-mndo 38234

This theorem is referenced by:  mndomgmid  38238  rngo1cl  38306