Theorem onnev 6490

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 6488	. . 3 ⊢ ¬ {{∅}} ∈ On
2		snex 5411	. . . 4 ⊢ {{∅}} ∈ V
3		id 23	. . . 4 ⊢ (On = V → On = V)
4	2, 3	eleqtrrid 2876	. . 3 ⊢ (On = V → {{∅}} ∈ On)
5	1, 4	mto 200	. 2 ⊢ ¬ On = V
6	5	neir 2967	1 ⊢ On ≠ V

Syntax hints:   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  Vcvv 3463  ∅c0 4294  {csn 4594  Oncon0 6361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-tr 5223  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-ord 6364  df-on 6365

This theorem is referenced by: (None)