Theorem onnev 6486

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 6484	. . 3 ⊢ ¬ {{∅}} ∈ On
2		snex 5404	. . . 4 ⊢ {{∅}} ∈ V
3		id 23	. . . 4 ⊢ (On = V → On = V)
4	2, 3	eleqtrrid 2867	. . 3 ⊢ (On = V → {{∅}} ∈ On)
5	1, 4	mto 200	. 2 ⊢ ¬ On = V
6	5	neir 2958	1 ⊢ On ≠ V

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  Vcvv 3450  ∅c0 4279  {csn 4584  Oncon0 6357

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6360  df-on 6361

This theorem is used by: (None)