Theorem mndoisexid 33668

Description: A monoid has an identity element. (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.)

Assertion

Ref	Expression
mndoisexid	|- ( G e. MndOp -> G e. ExId )

Proof of Theorem mndoisexid

Step	Hyp	Ref	Expression
1		elinel2 3800	. 2 |- ( G e. ( SemiGrp i^i ExId ) -> G e. ExId )
2		df-mndo 33666	. 2 |- MndOp = ( SemiGrp i^i ExId )
3	1, 2	eleq2s 2719	1 |- ( G e. MndOp -> G e. ExId )

Syntax hints:   -> wi 4   e. wcel 1990   i^i cin 3573   ExId cexid 33643   SemiGrpcsem 33659  MndOpcmndo 33665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-mndo 33666

This theorem is referenced by:  mndomgmid  33670  rngo1cl  33738