Theorem rint0 4274

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 4236	. . 3 |- ( X = (/) -> |^| X = |^| (/) )
2	1	ineq2d 3633	. 2 |- ( X = (/) -> ( A i^i |^| X ) = ( A i^i |^| (/) ) )
3		int0 4247	. . . 4 |- |^| (/) = _V
4	3	ineq2i 3630	. . 3 |- ( A i^i |^| (/) ) = ( A i^i _V )
5		inv1 3760	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2472	. 2 |- ( A i^i |^| (/) ) = A
7	2, 6	syl6eq 2500	1 |- ( X = (/) -> ( A i^i |^| X ) = A )

Syntax hints:   -> wi 4   = wceq 1443   _Vcvv 3044   i^i cin 3402   (/)c0 3730   |^|cint 4233

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ral 2741  df-v 3046  df-dif 3406  df-in 3410  df-ss 3417  df-nul 3731  df-int 4234

This theorem is referenced by:  incexclem  13887  incexc  13888  mrerintcl  15496  ismred2  15502  txtube  20648