Theorem riin0 3988

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 3930	. . 3 |- ( X = (/) -> |^|_ x e. X S = |^|_ x e. (/) S )
2	1	ineq2d 3364	. 2 |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = ( A i^i |^|_ x e. (/) S ) )
3		0iin 3975	. . . 4 |- |^|_ x e. (/) S = _V
4	3	ineq2i 3361	. . 3 |- ( A i^i |^|_ x e. (/) S ) = ( A i^i _V )
5		inv1 3487	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2217	. 2 |- ( A i^i |^|_ x e. (/) S ) = A
7	2, 6	eqtrdi 2245	1 |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = A )

Syntax hints:   -> wi 4   = wceq 1364   _Vcvv 2763   i^i cin 3156   (/)c0 3450   |^|_ciin 3917

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451  df-iin 3919

This theorem is referenced by: (None)