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Theorem onxpdisj 4769
Description: Ordinal numbers and ordered pairs are disjoint collections. This theorem can be used if we want to extend a set of ordinal numbers or ordered pairs with disjoint elements. See also snsn0non 4511. (Contributed by NM, 1-Jun-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
onxpdisj  |-  ( On 
i^i  ( _V  X.  _V ) )  =  (/)

Proof of Theorem onxpdisj
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 disj 3495 . 2  |-  ( ( On  i^i  ( _V 
X.  _V ) )  =  (/) 
<-> 
A. x  e.  On  -.  x  e.  ( _V  X.  _V ) )
2 on0eqel 4510 . . 3  |-  ( x  e.  On  ->  (
x  =  (/)  \/  (/)  e.  x
) )
3 0nelxp 4717 . . . . 5  |-  -.  (/)  e.  ( _V  X.  _V )
4 eleq1 2343 . . . . 5  |-  ( x  =  (/)  ->  ( x  e.  ( _V  X.  _V )  <->  (/)  e.  ( _V 
X.  _V ) ) )
53, 4mtbiri 294 . . . 4  |-  ( x  =  (/)  ->  -.  x  e.  ( _V  X.  _V ) )
6 0nelelxp 4718 . . . . 5  |-  ( x  e.  ( _V  X.  _V )  ->  -.  (/)  e.  x
)
76con2i 112 . . . 4  |-  ( (/)  e.  x  ->  -.  x  e.  ( _V  X.  _V ) )
85, 7jaoi 368 . . 3  |-  ( ( x  =  (/)  \/  (/)  e.  x
)  ->  -.  x  e.  ( _V  X.  _V ) )
92, 8syl 15 . 2  |-  ( x  e.  On  ->  -.  x  e.  ( _V  X.  _V ) )
101, 9mprgbir 2613 1  |-  ( On 
i^i  ( _V  X.  _V ) )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   (/)c0 3455   Oncon0 4392    X. cxp 4687
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-tr 4114  df-eprel 4305  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695
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