Theorem mndoisexid 39006

Description: Obsolete theorem, use mndid 18965 instead. A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
mndoisexid	⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId )

Proof of Theorem mndoisexid

Step	Hyp	Ref	Expression
1		elinel2 4102	. 2 ⊢ (𝐺 ∈ (SemiGrp ∩ ExId ) → 𝐺 ∈ ExId )
2		df-mndo 39004	. 2 ⊢ MndOp = (SemiGrp ∩ ExId )
3	1, 2	eleq2s 2882	1 ⊢ (𝐺 ∈ MndOp → 𝐺 ∈ ExId )

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147   ∩ cin 4092   ExId cexid 38981  SemiGrpcsem 38997  MndOpcmndo 39003

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-in 4094  df-mndo 39004

This theorem is used by:  mndomgmid  39008  rngo1cl  39076