Theorem rint0 4517

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 4478	. . 3 |- ( X = (/) -> |^| X = |^| (/) )
2	1	ineq2d 3814	. 2 |- ( X = (/) -> ( A i^i |^| X ) = ( A i^i |^| (/) ) )
3		int0 4490	. . . 4 |- |^| (/) = _V
4	3	ineq2i 3811	. . 3 |- ( A i^i |^| (/) ) = ( A i^i _V )
5		inv1 3970	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2644	. 2 |- ( A i^i |^| (/) ) = A
7	2, 6	syl6eq 2672	1 |- ( X = (/) -> ( A i^i |^| X ) = A )

Syntax hints:   -> wi 4   = wceq 1483   _Vcvv 3200   i^i cin 3573   (/)c0 3915   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-int 4476

This theorem is referenced by:  incexclem  14568  incexc  14569  mrerintcl  16257  ismred2  16263  txtube  21443  bj-mooreset  33056  bj-ismoored0  33061  bj-ismooredr2  33065