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Theorem grpon0 21821
Description: The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (New usage is discouraged.)
Hypothesis
Ref Expression
grpfo.1  |-  X  =  ran  G
Assertion
Ref Expression
grpon0  |-  ( G  e.  GrpOp  ->  X  =/=  (/) )

Proof of Theorem grpon0
Dummy variables  x  y  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpfo.1 . . 3  |-  X  =  ran  G
21grpolidinv 21820 . 2  |-  ( G  e.  GrpOp  ->  E. u  e.  X  A. x  e.  X  ( (
u G x )  =  x  /\  E. y  e.  X  (
y G x )  =  u ) )
3 rexn0 3754 . 2  |-  ( E. u  e.  X  A. x  e.  X  (
( u G x )  =  x  /\  E. y  e.  X  ( y G x )  =  u )  ->  X  =/=  (/) )
42, 3syl 16 1  |-  ( G  e.  GrpOp  ->  X  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605   A.wral 2711   E.wrex 2712   (/)c0 3613   ran crn 4908  (class class class)co 6110   GrpOpcgr 21805
This theorem is referenced by:  0ngrp  21830  rngon0  22035  vcoprnelem  22088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fo 5489  df-fv 5491  df-ov 6113  df-grpo 21810
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