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Theorem riin0 4165
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 4108 . . 3  |-  ( X  =  (/)  ->  |^|_ x  e.  X  S  =  |^|_
x  e.  (/)  S )
21ineq2d 3543 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  ( A  i^i  |^|_ x  e.  (/)  S ) )
3 0iin 4150 . . . 4  |-  |^|_ x  e.  (/)  S  =  _V
43ineq2i 3540 . . 3  |-  ( A  i^i  |^|_ x  e.  (/)  S )  =  ( A  i^i  _V )
5 inv1 3655 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2457 . 2  |-  ( A  i^i  |^|_ x  e.  (/)  S )  =  A
72, 6syl6eq 2485 1  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653   _Vcvv 2957    i^i cin 3320   (/)c0 3629   |^|_ciin 4095
This theorem is referenced by:  riinrab  4167  riiner  6978  mreriincl  13824  riinopn  16982  riincld  17109  fnemeet2  26397  pmapglb2N  30569  pmapglb2xN  30570
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ral 2711  df-v 2959  df-dif 3324  df-in 3328  df-ss 3335  df-nul 3630  df-iin 4097
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