Theorem rint0 4269

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 4232	. . 3 |- ( X = (/) -> |^| X = |^| (/) )
2	1	ineq2d 3653	. 2 |- ( X = (/) -> ( A i^i |^| X ) = ( A i^i |^| (/) ) )
3		int0 4243	. . . 4 |- |^| (/) = _V
4	3	ineq2i 3650	. . 3 |- ( A i^i |^| (/) ) = ( A i^i _V )
5		inv1 3765	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2480	. 2 |- ( A i^i |^| (/) ) = A
7	2, 6	syl6eq 2508	1 |- ( X = (/) -> ( A i^i |^| X ) = A )

Syntax hints:   -> wi 4   = wceq 1370   _Vcvv 3071   i^i cin 3428   (/)c0 3738   |^|cint 4229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-v 3073  df-dif 3432  df-in 3436  df-ss 3443  df-nul 3739  df-int 4230

This theorem is referenced by:  incexclem  13410  incexc  13411  mrerintcl  14646  ismred2  14652  txtube  19338