MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  grpn0 Structured version   Unicode version

Theorem grpn0 15569
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 2442 . . 3  |-  ( Base `  G )  =  (
Base `  G )
21grpbn0 15566 . 2  |-  ( G  e.  Grp  ->  ( Base `  G )  =/=  (/) )
3 fveq2 5690 . . . 4  |-  ( G  =  (/)  ->  ( Base `  G )  =  (
Base `  (/) ) )
4 base0 14212 . . . 4  |-  (/)  =  (
Base `  (/) )
53, 4syl6eqr 2492 . . 3  |-  ( G  =  (/)  ->  ( Base `  G )  =  (/) )
65necon3i 2649 . 2  |-  ( (
Base `  G )  =/=  (/)  ->  G  =/=  (/) )
72, 6syl 16 1  |-  ( G  e.  Grp  ->  G  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756    =/= wne 2605   (/)c0 3636   ` cfv 5417   Basecbs 14173   Grpcgrp 15409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5380  df-fun 5419  df-fv 5425  df-riota 6051  df-ov 6093  df-slot 14177  df-base 14178  df-0g 14379  df-mnd 15414  df-grp 15544
This theorem is referenced by:  lactghmga  15908
  Copyright terms: Public domain W3C validator