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Theorem onnev 4950
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
StepHypRef Expression
1 snsn0non 4692 . 2  |-  -.  { { (/) } }  e.  On
2 snex 4397 . . . 4  |-  { { (/)
} }  e.  _V
3 id 20 . . . 4  |-  ( On  =  _V  ->  On  =  _V )
42, 3syl5eleqr 2522 . . 3  |-  ( On  =  _V  ->  { { (/)
} }  e.  On )
54necon3bi 2639 . 2  |-  ( -. 
{ { (/) } }  e.  On  ->  On  =/=  _V )
61, 5ax-mp 8 1  |-  On  =/=  _V
Colors of variables: wff set class
Syntax hints:   -. wn 3    = wceq 1652    e. wcel 1725    =/= wne 2598   _Vcvv 2948   (/)c0 3620   {csn 3806   Oncon0 4573
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
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