Theorem onnev 6513

Description: The class of ordinal numbers is not equal to the universe. (Contributed by NM, 16-Jun-2007.) (Proof shortened by Mario Carneiro, 10-Jan-2013.) (Proof shortened by Wolf Lammen, 27-May-2024.)

Assertion

Ref	Expression
onnev	⊢ On ≠ V

Proof of Theorem onnev

Step	Hyp	Ref	Expression
1		snsn0non 6511	. . 3 ⊢ ¬ {{∅}} ∈ On
2		snex 5442	. . . 4 ⊢ {{∅}} ∈ V
3		id 22	. . . 4 ⊢ (On = V → On = V)
4	2, 3	eleqtrrid 2846	. . 3 ⊢ (On = V → {{∅}} ∈ On)
5	1, 4	mto 197	. 2 ⊢ ¬ On = V
6	5	neir 2941	1 ⊢ On ≠ V

Syntax hints:   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  Vcvv 3478  ∅c0 4339  {csn 4631  Oncon0 6386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390

This theorem is referenced by: (None)