Theorem rint0 4270

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 3643	. 2 |- ( X = (/) -> ( A i^i |^| X ) = ( A i^i |^| (/) ) )
3		int0 4243	. . . 4 |- |^| (/) = _V
4	3	ineq2i 3640	. . 3 |- ( A i^i |^| (/) ) = ( A i^i _V )
5		inv1 3768	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2433	. 2 |- ( A i^i |^| (/) ) = A
7	2, 6	syl6eq 2461	1 |- ( X = (/) -> ( A i^i |^| X ) = A )

Syntax hints:   -> wi 4   = wceq 1407   _Vcvv 3061   i^i cin 3415   (/)c0 3740   |^|cint 4229

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382

This theorem depends on definitions:  df-bi 187  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761  df-v 3063  df-dif 3419  df-in 3423  df-ss 3430  df-nul 3741  df-int 4230

This theorem is referenced by:  incexclem  13801  incexc  13802  mrerintcl  15213  ismred2  15219  txtube  20435