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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 3817 . 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 4881 . . . . 5  |-  -.  (/)  e.  ( _V  X.  _V )
4 eleq1 2528 . . . . 5  |-  ( x  =  (/)  ->  ( x  e.  ( _V  X.  _V )  <->  (/)  e.  ( _V 
X.  _V ) ) )
53, 4mtbiri 309 . . . 4  |-  ( x  =  (/)  ->  -.  x  e.  ( _V  X.  _V ) )
6 0nelelxp 4882 . . . . 5  |-  ( x  e.  ( _V  X.  _V )  ->  -.  (/)  e.  x
)
76con2i 125 . . . 4  |-  ( (/)  e.  x  ->  -.  x  e.  ( _V  X.  _V ) )
85, 7jaoi 385 . . 3  |-  ( ( x  =  (/)  \/  (/)  e.  x
)  ->  -.  x  e.  ( _V  X.  _V ) )
92, 8syl 17 . 2  |-  ( x  e.  On  ->  -.  x  e.  ( _V  X.  _V ) )
101, 9mprgbir 2764 1  |-  ( On 
i^i  ( _V  X.  _V ) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 374    = wceq 1455    e. wcel 1898   _Vcvv 3057    i^i cin 3415   (/)c0 3743    X. cxp 4851   Oncon0 5442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-tr 4512  df-eprel 4764  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-ord 5445  df-on 5446
This theorem is referenced by: (None)
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