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Theorem onnev 5550
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 5548 . 2  |-  -.  { { (/) } }  e.  On
2 snex 4641 . . . 4  |-  { { (/)
} }  e.  _V
3 id 22 . . . 4  |-  ( On  =  _V  ->  On  =  _V )
42, 3syl5eleqr 2556 . . 3  |-  ( On  =  _V  ->  { { (/)
} }  e.  On )
54necon3bi 2669 . 2  |-  ( -. 
{ { (/) } }  e.  On  ->  On  =/=  _V )
61, 5ax-mp 5 1  |-  On  =/=  _V
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    = wceq 1452    e. wcel 1904    =/= wne 2641   _Vcvv 3031   (/)c0 3722   {csn 3959   Oncon0 5430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-ord 5433  df-on 5434
This theorem is referenced by: (None)
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