MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  0ngrp Structured version   Visualization version   GIF version

Theorem 0ngrp 28215
Description: The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
0ngrp ¬ ∅ ∈ GrpOp

Proof of Theorem 0ngrp
StepHypRef Expression
1 neirr 3022 . 2 ¬ ∅ ≠ ∅
2 rn0 5789 . . . 4 ran ∅ = ∅
32eqcomi 2827 . . 3 ∅ = ran ∅
43grpon0 28206 . 2 (∅ ∈ GrpOp → ∅ ≠ ∅)
51, 4mto 198 1 ¬ ∅ ∈ GrpOp
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2105  wne 3013  c0 4288  ran crn 5549  GrpOpcgr 28193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-ov 7148  df-grpo 28197
This theorem is referenced by:  vsfval  28337  zrdivrng  35112
  Copyright terms: Public domain W3C validator