Theorem riin0 4082

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 4024	. . 3 |- ( X = (/) -> |^|_ x e. X S = |^|_ x e. (/) S )
2	1	ineq2d 3432	. 2 |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = ( A i^i |^|_ x e. (/) S ) )
3		0iin 4069	. . . 4 |- |^|_ x e. (/) S = _V
4	3	ineq2i 3429	. . 3 |- ( A i^i |^|_ x e. (/) S ) = ( A i^i _V )
5		inv1 3559	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2259	. 2 |- ( A i^i |^|_ x e. (/) S ) = A
7	2, 6	eqtrdi 2287	1 |- ( X = (/) -> ( A i^i |^|_ x e. X S ) = A )

Syntax hints:   -> wi 4   = wceq 1402   _Vcvv 2821   i^i cin 3219   (/)c0 3520   |^|_ciin 4011

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-dif 3222  df-in 3226  df-ss 3233  df-nul 3521  df-iin 4013

This theorem is referenced by: (None)