Theorem rint0 3938

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 3902	. . 3 |- ( X = (/) -> |^| X = |^| (/) )
2	1	ineq2d 3382	. 2 |- ( X = (/) -> ( A i^i |^| X ) = ( A i^i |^| (/) ) )
3		int0 3913	. . . 4 |- |^| (/) = _V
4	3	ineq2i 3379	. . 3 |- ( A i^i |^| (/) ) = ( A i^i _V )
5		inv1 3505	. . 3 |- ( A i^i _V ) = A
6	4, 5	eqtri 2228	. 2 |- ( A i^i |^| (/) ) = A
7	2, 6	eqtrdi 2256	1 |- ( X = (/) -> ( A i^i |^| X ) = A )

Syntax hints:   -> wi 4   = wceq 1373   _Vcvv 2776   i^i cin 3173   (/)c0 3468   |^|cint 3899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778  df-dif 3176  df-in 3180  df-ss 3187  df-nul 3469  df-int 3900

This theorem is referenced by: (None)