Theorem onnevOLD 6490

Description: Obsolete version of onnev 6489 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 6487	. 2 ⊢ ¬ {{∅}} ∈ On
2		snex 5425	. . . 4 ⊢ {{∅}} ∈ V
3		id 22	. . . 4 ⊢ (On = V → On = V)
4	2, 3	eleqtrrid 2832	. . 3 ⊢ (On = V → {{∅}} ∈ On)
5	4	necon3bi 2957	. 2 ⊢ (¬ {{∅}} ∈ On → On ≠ V)
6	1, 5	ax-mp 5	1 ⊢ On ≠ V

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2098   ≠ wne 2930  Vcvv 3463  ∅c0 4316  {csn 4622  Oncon0 6362

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 2696  ax-sep 5292  ax-nul 5299  ax-pr 5421

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 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3958  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5574  df-po 5582  df-so 5583  df-fr 5625  df-we 5627  df-ord 6365  df-on 6366

This theorem is referenced by: (None)