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Theorem onnev 5886
 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.)
Assertion
Ref Expression
onnev On ≠ V

Proof of Theorem onnev
StepHypRef Expression
1 snsn0non 5884 . 2 ¬ {{∅}} ∈ On
2 snex 4938 . . . 4 {{∅}} ∈ V
3 id 22 . . . 4 (On = V → On = V)
42, 3syl5eleqr 2737 . . 3 (On = V → {{∅}} ∈ On)
54necon3bi 2849 . 2 (¬ {{∅}} ∈ On → On ≠ V)
61, 5ax-mp 5 1 On ≠ V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  Vcvv 3231  ∅c0 3948  {csn 4210  Oncon0 5761 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-ord 5764  df-on 5765 This theorem is referenced by: (None)
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