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Theorem grpn0 16406
Description: A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)
Assertion
Ref Expression
grpn0  |-  ( G  e.  Grp  ->  G  =/=  (/) )

Proof of Theorem grpn0
StepHypRef Expression
1 eqid 2402 . . 3  |-  ( Base `  G )  =  (
Base `  G )
21grpbn0 16403 . 2  |-  ( G  e.  Grp  ->  ( Base `  G )  =/=  (/) )
3 fveq2 5849 . . . 4  |-  ( G  =  (/)  ->  ( Base `  G )  =  (
Base `  (/) ) )
4 base0 14882 . . . 4  |-  (/)  =  (
Base `  (/) )
53, 4syl6eqr 2461 . . 3  |-  ( G  =  (/)  ->  ( Base `  G )  =  (/) )
65necon3i 2643 . 2  |-  ( (
Base `  G )  =/=  (/)  ->  G  =/=  (/) )
72, 6syl 17 1  |-  ( G  e.  Grp  ->  G  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405    e. wcel 1842    =/= wne 2598   (/)c0 3738   ` cfv 5569   Basecbs 14841   Grpcgrp 16377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-iota 5533  df-fun 5571  df-fv 5577  df-riota 6240  df-ov 6281  df-slot 14845  df-base 14846  df-0g 15056  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-grp 16381
This theorem is referenced by:  lactghmga  16753
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