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Theorem onnevOLD 6288
 Description: Obsolete version of onnev 6287 as of 27-May-2024. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
onnevOLD On ≠ V

Proof of Theorem onnevOLD
StepHypRef Expression
1 snsn0non 6285 . 2 ¬ {{∅}} ∈ On
2 snex 5301 . . . 4 {{∅}} ∈ V
3 id 22 . . . 4 (On = V → On = V)
42, 3eleqtrrid 2897 . . 3 (On = V → {{∅}} ∈ On)
54necon3bi 3013 . 2 (¬ {{∅}} ∈ On → On ≠ V)
61, 5ax-mp 5 1 On ≠ V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  Vcvv 3442  ∅c0 4246  {csn 4528  Oncon0 6166 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-tr 5141  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6169  df-on 6170 This theorem is referenced by: (None)
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