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Theorem onxpdisj 4949
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 4692. (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 3660 . 2  |-  ( ( On  i^i  ( _V 
X.  _V ) )  =  (/) 
<-> 
A. x  e.  On  -.  x  e.  ( _V  X.  _V ) )
2 on0eqel 4691 . . 3  |-  ( x  e.  On  ->  (
x  =  (/)  \/  (/)  e.  x
) )
3 0nelxp 4898 . . . . 5  |-  -.  (/)  e.  ( _V  X.  _V )
4 eleq1 2495 . . . . 5  |-  ( x  =  (/)  ->  ( x  e.  ( _V  X.  _V )  <->  (/)  e.  ( _V 
X.  _V ) ) )
53, 4mtbiri 295 . . . 4  |-  ( x  =  (/)  ->  -.  x  e.  ( _V  X.  _V ) )
6 0nelelxp 4899 . . . . 5  |-  ( x  e.  ( _V  X.  _V )  ->  -.  (/)  e.  x
)
76con2i 114 . . . 4  |-  ( (/)  e.  x  ->  -.  x  e.  ( _V  X.  _V ) )
85, 7jaoi 369 . . 3  |-  ( ( x  =  (/)  \/  (/)  e.  x
)  ->  -.  x  e.  ( _V  X.  _V ) )
92, 8syl 16 . 2  |-  ( x  e.  On  ->  -.  x  e.  ( _V  X.  _V ) )
101, 9mprgbir 2768 1  |-  ( On 
i^i  ( _V  X.  _V ) )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 358    = wceq 1652    e. wcel 1725   _Vcvv 2948    i^i cin 3311   (/)c0 3620   Oncon0 4573    X. cxp 4868
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-xp 4876
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