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Theorem onnev 3232
Description: The class of ordinal numbers is not equal to the universe.
Assertion
Ref Expression
onnev |- On =/= V
Proof of Theorem onnev
StepHypRef Expression
1 0ex 2701 . . . 4 |- (/) e. V
2 opelxpi 3207 . . . 4 |- (((/) e. V /\ (/) e. V) -> <.(/), (/)>. e. (V X. V))
31, 1, 2mp2an 695 . . 3 |- <.(/), (/)>. e. (V X. V)
4 ne0i 2276 . . 3 |- (<.(/), (/)>. e. (V X. V) -> (V X. V) =/= (/))
53, 4ax-mp 7 . 2 |- (V X. V) =/= (/)
6 ineq1 2200 . . . . 5 |- (On = V -> (On i^i (V X. V)) = (V i^i (V X. V)))
7 onxpdisj 3231 . . . . 5 |- (On i^i (V X. V)) = (/)
8 incom 2198 . . . . . 6 |- (V i^i (V X. V)) = ((V X. V) i^i V)
9 inv1 2289 . . . . . 6 |- ((V X. V) i^i V) = (V X. V)
108, 9eqtr 1487 . . . . 5 |- (V i^i (V X. V)) = (V X. V)
116, 7, 103eqtr3g 1522 . . . 4 |- (On = V -> (/) = (V X. V))
1211eqcomd 1472 . . 3 |- (On = V -> (V X. V) = (/))
1312necon3i 1597 . 2 |- ((V X. V) =/= (/) -> On =/= V)
145, 13ax-mp 7 1 |- On =/= V
Colors of variables: wff set class
Syntax hints:   = wceq 953   e. wcel 955   =/= wne 1577  Vcvv 1802   i^i cin 2036  (/)c0 2270  <.cop 2401  Oncon0 2938   X. cxp 3158
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-xp 3174
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