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Theorem subgid 14875
Description: A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypothesis
Ref Expression
issubg.b  |-  B  =  ( Base `  G
)
Assertion
Ref Expression
subgid  |-  ( G  e.  Grp  ->  B  e.  (SubGrp `  G )
)

Proof of Theorem subgid
StepHypRef Expression
1 id 20 . 2  |-  ( G  e.  Grp  ->  G  e.  Grp )
2 ssid 3312 . . 3  |-  B  C_  B
32a1i 11 . 2  |-  ( G  e.  Grp  ->  B  C_  B )
4 issubg.b . . . 4  |-  B  =  ( Base `  G
)
54ressid 13453 . . 3  |-  ( G  e.  Grp  ->  ( Gs  B )  =  G )
65, 1eqeltrd 2463 . 2  |-  ( G  e.  Grp  ->  ( Gs  B )  e.  Grp )
74issubg 14873 . 2  |-  ( B  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  B  C_  B  /\  ( Gs  B )  e.  Grp ) )
81, 3, 6, 7syl3anbrc 1138 1  |-  ( G  e.  Grp  ->  B  e.  (SubGrp `  G )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    C_ wss 3265   ` cfv 5396  (class class class)co 6022   Basecbs 13398   ↾s cress 13399   Grpcgrp 14614  SubGrpcsubg 14867
This theorem is referenced by:  nsgid  14915  gaid2  15009  pgpfac1  15567  pgpfac  15571  ablfaclem2  15573  ablfac  15575
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-ress 13405  df-subg 14870
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