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Theorem riin0 3728
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 3671 . . 3  |-  ( X  =  (/)  ->  |^|_ x  e.  X  S  =  |^|_
x  e.  (/)  S )
21ineq2d 3138 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  ( A  i^i  |^|_ x  e.  (/)  S ) )
3 0iin 3715 . . . 4  |-  |^|_ x  e.  (/)  S  =  _V
43ineq2i 3135 . . 3  |-  ( A  i^i  |^|_ x  e.  (/)  S )  =  ( A  i^i  _V )
5 inv1 3253 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2060 . 2  |-  ( A  i^i  |^|_ x  e.  (/)  S )  =  A
72, 6syl6eq 2088 1  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243   _Vcvv 2557    i^i cin 2916   (/)c0 3224   |^|_ciin 3658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225  df-iin 3660
This theorem is referenced by: (None)
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