Theorem onnevOLD 6335

Description: Obsolete version of onnev 6334 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 6332	. 2 ⊢ ¬ {{∅}} ∈ On
2		snex 5324	. . . 4 ⊢ {{∅}} ∈ V
3		id 22	. . . 4 ⊢ (On = V → On = V)
4	2, 3	eleqtrrid 2845	. . 3 ⊢ (On = V → {{∅}} ∈ On)
5	4	necon3bi 2967	. 2 ⊢ (¬ {{∅}} ∈ On → On ≠ V)
6	1, 5	ax-mp 5	1 ⊢ On ≠ V

Syntax hints:  ¬ wn 3   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  Vcvv 3408  ∅c0 4237  {csn 4541  Oncon0 6213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-tr 5162  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-ord 6216  df-on 6217

This theorem is referenced by: (None)