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Theorem onxpdisj 5089
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 5002. (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 3872 . 2  |-  ( ( On  i^i  ( _V 
X.  _V ) )  =  (/) 
<-> 
A. x  e.  On  -.  x  e.  ( _V  X.  _V ) )
2 on0eqel 5001 . . 3  |-  ( x  e.  On  ->  (
x  =  (/)  \/  (/)  e.  x
) )
3 0nelxp 5033 . . . . 5  |-  -.  (/)  e.  ( _V  X.  _V )
4 eleq1 2539 . . . . 5  |-  ( x  =  (/)  ->  ( x  e.  ( _V  X.  _V )  <->  (/)  e.  ( _V 
X.  _V ) ) )
53, 4mtbiri 303 . . . 4  |-  ( x  =  (/)  ->  -.  x  e.  ( _V  X.  _V ) )
6 0nelelxp 5034 . . . . 5  |-  ( x  e.  ( _V  X.  _V )  ->  -.  (/)  e.  x
)
76con2i 120 . . . 4  |-  ( (/)  e.  x  ->  -.  x  e.  ( _V  X.  _V ) )
85, 7jaoi 379 . . 3  |-  ( ( x  =  (/)  \/  (/)  e.  x
)  ->  -.  x  e.  ( _V  X.  _V ) )
92, 8syl 16 . 2  |-  ( x  e.  On  ->  -.  x  e.  ( _V  X.  _V ) )
101, 9mprgbir 2831 1  |-  ( On 
i^i  ( _V  X.  _V ) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    = wceq 1379    e. wcel 1767   _Vcvv 3118    i^i cin 3480   (/)c0 3790   Oncon0 4884    X. cxp 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-tr 4547  df-eprel 4797  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-xp 5011
This theorem is referenced by: (None)
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