Theorem rint0 4294

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 4256	. . 3 |- ( X = (/) -> |^| X = |^| (/) )
2	1	ineq2d 3665	. 2 |- ( X = (/) -> ( A i^i |^| X ) = ( A i^i |^| (/) ) )
3		int0 4267	. . . 4 |- |^| (/) = _V
4	3	ineq2i 3662	. . 3 |- ( A i^i |^| (/) ) = ( A i^i _V )
5		inv1 3790	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2452	. 2 |- ( A i^i |^| (/) ) = A
7	2, 6	syl6eq 2480	1 |- ( X = (/) -> ( A i^i |^| X ) = A )

Syntax hints:   -> wi 4   = wceq 1438   _Vcvv 3082   i^i cin 3436   (/)c0 3762   |^|cint 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ral 2781  df-v 3084  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3763  df-int 4254

This theorem is referenced by:  incexclem  13887  incexc  13888  mrerintcl  15496  ismred2  15502  txtube  20647