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Theorem 0ngrp 30540
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 2947 . 2 ¬ ∅ ≠ ∅
2 rn0 5939 . . . 4 ran ∅ = ∅
32eqcomi 2744 . . 3 ∅ = ran ∅
43grpon0 30531 . 2 (∅ ∈ GrpOp → ∅ ≠ ∅)
51, 4mto 197 1 ¬ ∅ ∈ GrpOp
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wcel 2106  wne 2938  c0 4339  ran crn 5690  GrpOpcgr 30518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-grpo 30522
This theorem is referenced by:  vsfval  30662  zrdivrng  37940
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