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Theorem subgid 16008
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 22 . 2  |-  ( G  e.  Grp  ->  G  e.  Grp )
2 ssid 3523 . . 3  |-  B  C_  B
32a1i 11 . 2  |-  ( G  e.  Grp  ->  B  C_  B )
4 issubg.b . . . 4  |-  B  =  ( Base `  G
)
54ressid 14550 . . 3  |-  ( G  e.  Grp  ->  ( Gs  B )  =  G )
65, 1eqeltrd 2555 . 2  |-  ( G  e.  Grp  ->  ( Gs  B )  e.  Grp )
74issubg 16006 . 2  |-  ( B  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  B  C_  B  /\  ( Gs  B )  e.  Grp ) )
81, 3, 6, 7syl3anbrc 1180 1  |-  ( G  e.  Grp  ->  B  e.  (SubGrp `  G )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    C_ wss 3476   ` cfv 5588  (class class class)co 6284   Basecbs 14490   ↾s cress 14491   Grpcgrp 15727  SubGrpcsubg 16000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-ress 14497  df-subg 16003
This theorem is referenced by:  nsgid  16052  gaid2  16146  pgpfac1  16933  pgpfac  16937  ablfaclem2  16939  ablfac  16941
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