Theorem onnev 6084

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.)

Assertion

Ref	Expression
onnev	⊢ On ≠ V

Proof of Theorem onnev

Step	Hyp	Ref	Expression
1		snsn0non 6082	. 2 ⊢ ¬ {{∅}} ∈ On
2		snex 5130	. . . 4 ⊢ {{∅}} ∈ V
3		id 22	. . . 4 ⊢ (On = V → On = V)
4	2, 3	syl5eleqr 2914	. . 3 ⊢ (On = V → {{∅}} ∈ On)
5	4	necon3bi 3026	. 2 ⊢ (¬ {{∅}} ∈ On → On ≠ V)
6	1, 5	ax-mp 5	1 ⊢ On ≠ V

Syntax hints:  ¬ wn 3   = wceq 1658   ∈ wcel 2166   ≠ wne 3000  Vcvv 3415  ∅c0 4145  {csn 4398  Oncon0 5964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-sbc 3664  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-pss 3815  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-tr 4977  df-eprel 5256  df-po 5264  df-so 5265  df-fr 5302  df-we 5304  df-ord 5967  df-on 5968

This theorem is referenced by: (None)