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Theorem 0ngrp 30258
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 2941 . 2 ¬ ∅ ≠ ∅
2 rn0 5916 . . . 4 ran ∅ = ∅
32eqcomi 2733 . . 3 ∅ = ran ∅
43grpon0 30249 . 2 (∅ ∈ GrpOp → ∅ ≠ ∅)
51, 4mto 196 1 ¬ ∅ ∈ GrpOp
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2098  wne 2932  c0 4315  ran crn 5668  GrpOpcgr 30236
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fo 6540  df-fv 6542  df-ov 7405  df-grpo 30240
This theorem is referenced by:  vsfval  30380  zrdivrng  37324
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