Theorem mndoisexid 26116

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		elin 3628	. . 3 |- ( G e. ( SemiGrp i^i ExId ) <-> ( G e. SemiGrp /\ G e. ExId ) )
2	1	simprbi 470	. 2 |- ( G e. ( SemiGrp i^i ExId ) -> G e. ExId )
3		df-mndo 26114	. 2 |- MndOp = ( SemiGrp i^i ExId )
4	2, 3	eleq2s 2557	1 |- ( G e. MndOp -> G e. ExId )

Syntax hints:   -> wi 4   e. wcel 1897   i^i cin 3414   ExId cexid 26090   SemiGrpcsem 26106  MndOpcmndo 26113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-in 3422  df-mndo 26114

This theorem is referenced by:  mndomgmid  26118  rngo1cl  26205