Theorem rint0 3923

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 3887	. . 3 |- ( X = (/) -> |^| X = |^| (/) )
2	1	ineq2d 3373	. 2 |- ( X = (/) -> ( A i^i |^| X ) = ( A i^i |^| (/) ) )
3		int0 3898	. . . 4 |- |^| (/) = _V
4	3	ineq2i 3370	. . 3 |- ( A i^i |^| (/) ) = ( A i^i _V )
5		inv1 3496	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2225	. 2 |- ( A i^i |^| (/) ) = A
7	2, 6	eqtrdi 2253	1 |- ( X = (/) -> ( A i^i |^| X ) = A )

Syntax hints:   -> wi 4   = wceq 1372   _Vcvv 2771   i^i cin 3164   (/)c0 3459   |^|cint 3884

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178  df-nul 3460  df-int 3885

This theorem is referenced by: (None)