Theorem rint0 3675

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 3639	. . 3 |- ( X = (/) -> |^| X = |^| (/) )
2	1	ineq2d 3167	. 2 |- ( X = (/) -> ( A i^i |^| X ) = ( A i^i |^| (/) ) )
3		int0 3650	. . . 4 |- |^| (/) = _V
4	3	ineq2i 3164	. . 3 |- ( A i^i |^| (/) ) = ( A i^i _V )
5		inv1 3280	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2101	. 2 |- ( A i^i |^| (/) ) = A
7	2, 6	syl6eq 2129	1 |- ( X = (/) -> ( A i^i |^| X ) = A )

Syntax hints:   -> wi 4   = wceq 1284   _Vcvv 2601   i^i cin 2972   (/)c0 3251   |^|cint 3636

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-dif 2975  df-in 2979  df-ss 2986  df-nul 3252  df-int 3637

This theorem is referenced by: (None)