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Theorem grpn0 15892
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 2467 . . 3  |-  ( Base `  G )  =  (
Base `  G )
21grpbn0 15889 . 2  |-  ( G  e.  Grp  ->  ( Base `  G )  =/=  (/) )
3 fveq2 5866 . . . 4  |-  ( G  =  (/)  ->  ( Base `  G )  =  (
Base `  (/) ) )
4 base0 14529 . . . 4  |-  (/)  =  (
Base `  (/) )
53, 4syl6eqr 2526 . . 3  |-  ( G  =  (/)  ->  ( Base `  G )  =  (/) )
65necon3i 2707 . 2  |-  ( (
Base `  G )  =/=  (/)  ->  G  =/=  (/) )
72, 6syl 16 1  |-  ( G  e.  Grp  ->  G  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    =/= wne 2662   (/)c0 3785   ` cfv 5588   Basecbs 14490   Grpcgrp 15727
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-reu 2821  df-rmo 2822  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-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-iota 5551  df-fun 5590  df-fv 5596  df-riota 6245  df-ov 6287  df-slot 14494  df-base 14495  df-0g 14697  df-mnd 15732  df-grp 15867
This theorem is referenced by:  lactghmga  16234
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