Theorem rint0 3885

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 3849	. . 3 |- ( X = (/) -> |^| X = |^| (/) )
2	1	ineq2d 3338	. 2 |- ( X = (/) -> ( A i^i |^| X ) = ( A i^i |^| (/) ) )
3		int0 3860	. . . 4 |- |^| (/) = _V
4	3	ineq2i 3335	. . 3 |- ( A i^i |^| (/) ) = ( A i^i _V )
5		inv1 3461	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2198	. 2 |- ( A i^i |^| (/) ) = A
7	2, 6	eqtrdi 2226	1 |- ( X = (/) -> ( A i^i |^| X ) = A )

Syntax hints:   -> wi 4   = wceq 1353   _Vcvv 2739   i^i cin 3130   (/)c0 3424   |^|cint 3846

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425  df-int 3847

This theorem is referenced by: (None)