Theorem onnevOLD 6284

Description: Obsolete version of onnev 6283 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

Step	Hyp	Ref	Expression
1		snsn0non 6281	. 2 ⊢ ¬ {{∅}} ∈ On
2		snex 5293	. . . 4 ⊢ {{∅}} ∈ V
3		id 22	. . . 4 ⊢ (On = V → On = V)
4	2, 3	eleqtrrid 2858	. . 3 ⊢ (On = V → {{∅}} ∈ On)
5	4	necon3bi 2975	. 2 ⊢ (¬ {{∅}} ∈ On → On ≠ V)
6	1, 5	ax-mp 5	1 ⊢ On ≠ V

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2112   ≠ wne 2949  Vcvv 3407  ∅c0 4221  {csn 4515  Oncon0 6162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pr 5291

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-ral 3073  df-rex 3074  df-rab 3077  df-v 3409  df-sbc 3694  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-op 4522  df-uni 4792  df-br 5026  df-opab 5088  df-tr 5132  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-we 5478  df-ord 6165  df-on 6166

This theorem is referenced by: (None)