Theorem onnevOLD 6502

Description: Obsolete version of onnev 6501 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 6499	. 2 ⊢ ¬ {{∅}} ∈ On
2		snex 5437	. . . 4 ⊢ {{∅}} ∈ V
3		id 22	. . . 4 ⊢ (On = V → On = V)
4	2, 3	eleqtrrid 2836	. . 3 ⊢ (On = V → {{∅}} ∈ On)
5	4	necon3bi 2964	. 2 ⊢ (¬ {{∅}} ∈ On → On ≠ V)
6	1, 5	ax-mp 5	1 ⊢ On ≠ V

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098   ≠ wne 2937  Vcvv 3473  ∅c0 4326  {csn 4632  Oncon0 6374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-tr 5270  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-ord 6377  df-on 6378

This theorem is referenced by: (None)