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Theorem onxpdisj 5561
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 5560. (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 3839 . 2  |-  ( ( On  i^i  ( _V 
X.  _V ) )  =  (/) 
<-> 
A. x  e.  On  -.  x  e.  ( _V  X.  _V ) )
2 on0eqel 5559 . . 3  |-  ( x  e.  On  ->  (
x  =  (/)  \/  (/)  e.  x
) )
3 0nelxp 4882 . . . . 5  |-  -.  (/)  e.  ( _V  X.  _V )
4 eleq1 2501 . . . . 5  |-  ( x  =  (/)  ->  ( x  e.  ( _V  X.  _V )  <->  (/)  e.  ( _V 
X.  _V ) ) )
53, 4mtbiri 304 . . . 4  |-  ( x  =  (/)  ->  -.  x  e.  ( _V  X.  _V ) )
6 0nelelxp 4883 . . . . 5  |-  ( x  e.  ( _V  X.  _V )  ->  -.  (/)  e.  x
)
76con2i 123 . . . 4  |-  ( (/)  e.  x  ->  -.  x  e.  ( _V  X.  _V ) )
85, 7jaoi 380 . . 3  |-  ( ( x  =  (/)  \/  (/)  e.  x
)  ->  -.  x  e.  ( _V  X.  _V ) )
92, 8syl 17 . 2  |-  ( x  e.  On  ->  -.  x  e.  ( _V  X.  _V ) )
101, 9mprgbir 2796 1  |-  ( On 
i^i  ( _V  X.  _V ) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 369    = wceq 1437    e. wcel 1870   _Vcvv 3087    i^i cin 3441   (/)c0 3767    X. cxp 4852   Oncon0 5442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-ord 5445  df-on 5446
This theorem is referenced by: (None)
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