Theorem onnev 6489

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 6487	. . 3 ⊢ ¬ {{∅}} ∈ On
2		snex 5410	. . . 4 ⊢ {{∅}} ∈ V
3		id 23	. . . 4 ⊢ (On = V → On = V)
4	2, 3	eleqtrrid 2870	. . 3 ⊢ (On = V → {{∅}} ∈ On)
5	1, 4	mto 200	. 2 ⊢ ¬ On = V
6	5	neir 2961	1 ⊢ On ≠ V

Syntax hints:   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ∅c0 4286  {csn 4589  Oncon0 6360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364

This theorem is referenced by: (None)