Theorem mndisexid 10387

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

Assertion

Ref	Expression
mndisexid	|- (G e. Mnd -> G e. ExId )

Proof of Theorem mndisexid

Step	Hyp	Ref	Expression
1		df-mnd 10385	. . 3 |- Mnd = (SemiGrp i^i ExId )
2	1	eleq2i 1961	. 2 |- (G e. Mnd <-> G e. (SemiGrp i^i ExId ))
3		elin 2786	. . 3 |- (G e. (SemiGrp i^i ExId ) <-> (G e. SemiGrp /\ G e. ExId ))
4	3	simprbi 353	. 2 |- (G e. (SemiGrp i^i ExId ) -> G e. ExId )
5	2, 4	sylbi 216	1 |- (G e. Mnd -> G e. ExId )

Syntax hints:   -> wi 3   e. wcel 1300   i^i cin 2592   ExId cexid 10361  SemiGrpcsem 10377  Mndcmnd 10384

This theorem is referenced by:  mndmgmid 10389  ismnd2 10392  ring1cl 10415  expus 14726

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-in 2603  df-mnd 10385

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