Theorem riin0 3937

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 3880	. . 3 |- ( X = (/) -> |^|_ x e. X S = |^|_ x e. (/) S )
2	1	ineq2d 3323	. 2 |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = ( A i^i |^|_ x e. (/) S ) )
3		0iin 3924	. . . 4 |- |^|_ x e. (/) S = _V
4	3	ineq2i 3320	. . 3 |- ( A i^i |^|_ x e. (/) S ) = ( A i^i _V )
5		inv1 3445	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2186	. 2 |- ( A i^i |^|_ x e. (/) S ) = A
7	2, 6	eqtrdi 2215	1 |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = A )

Syntax hints:   -> wi 4   = wceq 1343   _Vcvv 2726   i^i cin 3115   (/)c0 3409   |^|_ciin 3867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-dif 3118  df-in 3122  df-ss 3129  df-nul 3410  df-iin 3869

This theorem is referenced by: (None)