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Theorem riin0 4399
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 4340 . . 3  |-  ( X  =  (/)  ->  |^|_ x  e.  X  S  =  |^|_
x  e.  (/)  S )
21ineq2d 3700 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  ( A  i^i  |^|_ x  e.  (/)  S ) )
3 0iin 4383 . . . 4  |-  |^|_ x  e.  (/)  S  =  _V
43ineq2i 3697 . . 3  |-  ( A  i^i  |^|_ x  e.  (/)  S )  =  ( A  i^i  _V )
5 inv1 3812 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2496 . 2  |-  ( A  i^i  |^|_ x  e.  (/)  S )  =  A
72, 6syl6eq 2524 1  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   _Vcvv 3113    i^i cin 3475   (/)c0 3785   |^|_ciin 4326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786  df-iin 4328
This theorem is referenced by:  riinrab  4401  riiner  7384  mreriincl  14852  riinopn  19200  riincld  19327  fnemeet2  29804  pmapglb2N  34576  pmapglb2xN  34577
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