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Theorem subgid 16770
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 23 . 2  |-  ( G  e.  Grp  ->  G  e.  Grp )
2 ssid 3489 . . 3  |-  B  C_  B
32a1i 11 . 2  |-  ( G  e.  Grp  ->  B  C_  B )
4 issubg.b . . . 4  |-  B  =  ( Base `  G
)
54ressid 15146 . . 3  |-  ( G  e.  Grp  ->  ( Gs  B )  =  G )
65, 1eqeltrd 2517 . 2  |-  ( G  e.  Grp  ->  ( Gs  B )  e.  Grp )
74issubg 16768 . 2  |-  ( B  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  B  C_  B  /\  ( Gs  B )  e.  Grp ) )
81, 3, 6, 7syl3anbrc 1189 1  |-  ( G  e.  Grp  ->  B  e.  (SubGrp `  G )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870    C_ wss 3442   ` cfv 5601  (class class class)co 6305   Basecbs 15084   ↾s cress 15085   Grpcgrp 16620  SubGrpcsubg 16762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-ress 15091  df-subg 16765
This theorem is referenced by:  nsgid  16814  gaid2  16908  pgpfac1  17648  pgpfac  17652  ablfaclem2  17654  ablfac  17656  efghgrpOLD  25946
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