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Theorem riin0 4376
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 4317 . . 3  |-  ( X  =  (/)  ->  |^|_ x  e.  X  S  =  |^|_
x  e.  (/)  S )
21ineq2d 3670 . 2  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  ( A  i^i  |^|_ x  e.  (/)  S ) )
3 0iin 4360 . . . 4  |-  |^|_ x  e.  (/)  S  =  _V
43ineq2i 3667 . . 3  |-  ( A  i^i  |^|_ x  e.  (/)  S )  =  ( A  i^i  _V )
5 inv1 3795 . . 3  |-  ( A  i^i  _V )  =  A
64, 5eqtri 2458 . 2  |-  ( A  i^i  |^|_ x  e.  (/)  S )  =  A
72, 6syl6eq 2486 1  |-  ( X  =  (/)  ->  ( A  i^i  |^|_ x  e.  X  S )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   _Vcvv 3087    i^i cin 3441   (/)c0 3767   |^|_ciin 4303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ral 2787  df-v 3089  df-dif 3445  df-in 3449  df-ss 3456  df-nul 3768  df-iin 4305
This theorem is referenced by:  riinrab  4378  riiner  7444  mreriincl  15455  riinopn  19869  riincld  19990  fnemeet2  30808  pmapglb2N  33044  pmapglb2xN  33045
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