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Theorem subgid 14623
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 19 . 2  |-  ( G  e.  Grp  ->  G  e.  Grp )
2 ssid 3197 . . 3  |-  B  C_  B
32a1i 10 . 2  |-  ( G  e.  Grp  ->  B  C_  B )
4 issubg.b . . . 4  |-  B  =  ( Base `  G
)
54ressid 13203 . . 3  |-  ( G  e.  Grp  ->  ( Gs  B )  =  G )
65, 1eqeltrd 2357 . 2  |-  ( G  e.  Grp  ->  ( Gs  B )  e.  Grp )
74issubg 14621 . 2  |-  ( B  e.  (SubGrp `  G
)  <->  ( G  e. 
Grp  /\  B  C_  B  /\  ( Gs  B )  e.  Grp ) )
81, 3, 6, 7syl3anbrc 1136 1  |-  ( G  e.  Grp  ->  B  e.  (SubGrp `  G )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149   Grpcgrp 14362  SubGrpcsubg 14615
This theorem is referenced by:  nsgid  14663  gaid2  14757  pgpfac1  15315  pgpfac  15319  ablfaclem2  15321  ablfac  15323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-ress 13155  df-subg 14618
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