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Theorem 0ngrp 28290
 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 3027 . 2 ¬ ∅ ≠ ∅
2 rn0 5798 . . . 4 ran ∅ = ∅
32eqcomi 2832 . . 3 ∅ = ran ∅
43grpon0 28281 . 2 (∅ ∈ GrpOp → ∅ ≠ ∅)
51, 4mto 199 1 ¬ ∅ ∈ GrpOp
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∈ wcel 2114   ≠ wne 3018  ∅c0 4293  ran crn 5558  GrpOpcgr 28268 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fo 6363  df-fv 6365  df-ov 7161  df-grpo 28272 This theorem is referenced by:  vsfval  28412  zrdivrng  35233
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