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Theorem riin0 3942
Description: Relative intersection of an empty family. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Assertion
Ref Expression
riin0  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
Distinct variable groups:    x, A    x, X
Allowed substitution hint:    S( x)

Proof of Theorem riin0
StepHypRef Expression
1 iineq1 3885 . . 3  |-  ( X  =  (/)  ->  |^|_ x  e.  X  S  =  |^|_
x  e.  (/)  S )
21ineq2d 3328 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  ( A  i^i  |^|_ x  e.  (/)  S ) )
3 0iin 3929 . . . 4  |-  |^|_ x  e.  (/)  S  =  _V
43ineq2i 3325 . . 3  |-  ( A  i^i  |^|_ x  e.  (/)  S )  =  ( A  i^i  _V )
5 inv1 3450 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2191 . 2  |-  ( A  i^i  |^|_ x  e.  (/)  S )  =  A
72, 6eqtrdi 2219 1  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1348   _Vcvv 2730    i^i cin 3120   (/)c0 3414   |^|_ciin 3872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-iin 3874
This theorem is referenced by: (None)
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