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Theorem onnev 6283
 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 6281 . . 3 ¬ {{∅}} ∈ On
2 snex 5300 . . . 4 {{∅}} ∈ V
3 id 22 . . . 4 (On = V → On = V)
42, 3eleqtrrid 2900 . . 3 (On = V → {{∅}} ∈ On)
51, 4mto 200 . 2 ¬ On = V
65neir 2993 1 On ≠ V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  Vcvv 3444  ∅c0 4246  {csn 4528  Oncon0 6163 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166  df-on 6167 This theorem is referenced by: (None)
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