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Theorem ixpprc 6853
Description: A cartesian product of proper-class many sets is empty, because any function in the cartesian product has to be a set with domain  A, which is not possible for a proper class domain. (Contributed by Mario Carneiro, 25-Jan-2015.)
Assertion
Ref Expression
ixpprc  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem ixpprc
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 neq0 3478 . . 3  |-  ( -.  X_ x  e.  A  B  =  (/)  <->  E. f 
f  e.  X_ x  e.  A  B )
2 ixpfn 6838 . . . . 5  |-  ( f  e.  X_ x  e.  A  B  ->  f  Fn  A
)
3 fndm 5359 . . . . . 6  |-  ( f  Fn  A  ->  dom  f  =  A )
4 vex 2804 . . . . . . 7  |-  f  e. 
_V
54dmex 4957 . . . . . 6  |-  dom  f  e.  _V
63, 5syl6eqelr 2385 . . . . 5  |-  ( f  Fn  A  ->  A  e.  _V )
72, 6syl 15 . . . 4  |-  ( f  e.  X_ x  e.  A  B  ->  A  e.  _V )
87exlimiv 1624 . . 3  |-  ( E. f  f  e.  X_ x  e.  A  B  ->  A  e.  _V )
91, 8sylbi 187 . 2  |-  ( -.  X_ x  e.  A  B  =  (/)  ->  A  e.  _V )
109con1i 121 1  |-  ( -.  A  e.  _V  ->  X_ x  e.  A  B  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   dom cdm 4705    Fn wfn 5266   X_cixp 6833
This theorem is referenced by:  ixpexg  6856  ixpssmap2g  6861  ixpssmapg  6862  resixpfo  6870
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ixp 6834
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