Theorem mndoisexid 38190

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 4142	. 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ ExId )
2		df-mndo 38188	. 2 ⊢ MndOp = (SemiGrp ∩ ExId )
3	1, 2	eleq2s 2854	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId )

Syntax hints:   → wi 4   ∈ wcel 2114   ∩ cin 3888   ExId cexid 38165  SemiGrpcsem 38181  MndOpcmndo 38187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-in 3896  df-mndo 38188

This theorem is referenced by:  mndomgmid  38192  rngo1cl  38260