Theorem riin0 3970

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

Step	Hyp	Ref	Expression
1		iineq1 3912	. . 3 |- ( X = (/) -> |^|_ x e. X S = |^|_ x e. (/) S )
2	1	ineq2d 3348	. 2 |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = ( A i^i |^|_ x e. (/) S ) )
3		0iin 3957	. . . 4 |- |^|_ x e. (/) S = _V
4	3	ineq2i 3345	. . 3 |- ( A i^i |^|_ x e. (/) S ) = ( A i^i _V )
5		inv1 3471	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2208	. 2 |- ( A i^i |^|_ x e. (/) S ) = A
7	2, 6	eqtrdi 2236	1 |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = A )

Syntax hints:   -> wi 4   = wceq 1363   _Vcvv 2749   i^i cin 3140   (/)c0 3434   |^|_ciin 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-dif 3143  df-in 3147  df-ss 3154  df-nul 3435  df-iin 3901

This theorem is referenced by: (None)