Theorem rint0 3967

Description: Relative intersection of an empty set. (Contributed by Stefan O'Rear, 3-Apr-2015.)

Assertion

Ref	Expression
rint0	|- ( X = (/) -> ( A i^i |^| X ) = A )

Proof of Theorem rint0

Step	Hyp	Ref	Expression
1		inteq 3931	. . 3 |- ( X = (/) -> |^| X = |^| (/) )
2	1	ineq2d 3408	. 2 |- ( X = (/) -> ( A i^i |^| X ) = ( A i^i |^| (/) ) )
3		int0 3942	. . . 4 |- |^| (/) = _V
4	3	ineq2i 3405	. . 3 |- ( A i^i |^| (/) ) = ( A i^i _V )
5		inv1 3531	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2252	. 2 |- ( A i^i |^| (/) ) = A
7	2, 6	eqtrdi 2280	1 |- ( X = (/) -> ( A i^i |^| X ) = A )

Syntax hints:   -> wi 4   = wceq 1397   _Vcvv 2802   i^i cin 3199   (/)c0 3494   |^|cint 3928

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-int 3929

This theorem is referenced by: (None)