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Theorem subgid 14639
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 3210 . . 3  |-  B  C_  B
32a1i 10 . 2  |-  ( G  e.  Grp  ->  B  C_  B )
4 issubg.b . . . 4  |-  B  =  ( Base `  G
)
54ressid 13219 . . 3  |-  ( G  e.  Grp  ->  ( Gs  B )  =  G )
65, 1eqeltrd 2370 . 2  |-  ( G  e.  Grp  ->  ( Gs  B )  e.  Grp )
74issubg 14637 . 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 1632    e. wcel 1696    C_ wss 3165   ` cfv 5271  (class class class)co 5874   Basecbs 13164   ↾s cress 13165   Grpcgrp 14378  SubGrpcsubg 14631
This theorem is referenced by:  nsgid  14679  gaid2  14773  pgpfac1  15331  pgpfac  15335  ablfaclem2  15337  ablfac  15339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-ress 13171  df-subg 14634
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