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Theorem onnev 6482
Description: The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.) (Proof shortened by Wolf Lammen, 27-May-2024.)
Assertion
Ref Expression
onnev On ≠ V

Proof of Theorem onnev
StepHypRef Expression
1 snsn0non 6480 . . 3 ¬ {{∅}} ∈ On
2 snex 5422 . . . 4 {{∅}} ∈ V
3 id 22 . . . 4 (On = V → On = V)
42, 3eleqtrrid 2832 . . 3 (On = V → {{∅}} ∈ On)
51, 4mto 196 . 2 ¬ On = V
65neir 2935 1 On ≠ V
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  wcel 2098  wne 2932  Vcvv 3466  c0 4315  {csn 4621  Oncon0 6355
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-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-tr 5257  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-ord 6358  df-on 6359
This theorem is referenced by: (None)
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