Theorem rint0 4322

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 4285	. . 3 |- ( X = (/) -> |^| X = |^| (/) )
2	1	ineq2d 3700	. 2 |- ( X = (/) -> ( A i^i |^| X ) = ( A i^i |^| (/) ) )
3		int0 4296	. . . 4 |- |^| (/) = _V
4	3	ineq2i 3697	. . 3 |- ( A i^i |^| (/) ) = ( A i^i _V )
5		inv1 3812	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2496	. 2 |- ( A i^i |^| (/) ) = A
7	2, 6	syl6eq 2524	1 |- ( X = (/) -> ( A i^i |^| X ) = A )

Syntax hints:   -> wi 4   = wceq 1379   _Vcvv 3113   i^i cin 3475   (/)c0 3785   |^|cint 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786  df-int 4283

This theorem is referenced by:  incexclem  13607  incexc  13608  mrerintcl  14848  ismred2  14854  txtube  19876