Theorem iin0 2730

Description: An indexed intersection of the empty set, with a non-empty index set, is empty.

Assertion

Ref	Expression
iin0	|- (A =/= (/) <-> |^|_x e. A (/) = (/))

Distinct variable group:   x,A

Proof of Theorem iin0

Step	Hyp	Ref	Expression
1		r19.3rzv 2338	. . . 4 |- (A =/= (/) -> (y e. (/) <-> A.x e. A y e. (/)))
2	1	abbi2dv 1570	. . 3 |- (A =/= (/) -> (/) = {y | A.x e. A y e. (/)})
3		df-iin 2559	. . 3 |- |^|_x e. A (/) = {y | A.x e. A y e. (/)}
4	2, 3	syl6reqr 1518	. 2 |- (A =/= (/) -> |^|_x e. A (/) = (/))
5		0ex 2701	. . . . . 6 |- (/) e. V
6		n0i 2275	. . . . . 6 |- ((/) e. V -> -. V = (/))
7	5, 6	ax-mp 7	. . . . 5 |- -. V = (/)
8		0iin 2596	. . . . . 6 |- |^|_x e. (/) (/) = V
9	8	eqeq1i 1474	. . . . 5 |- (|^|_x e. (/) (/) = (/) <-> V = (/))
10	7, 9	mtbir 192	. . . 4 |- -. |^|_x e. (/) (/) = (/)
11		iineq1 2566	. . . . 5 |- (A = (/) -> |^|_x e. A (/) = |^|_x e. (/) (/))
12	11	eqeq1d 1475	. . . 4 |- (A = (/) -> (|^|_x e. A (/) = (/) <-> |^|_x e. (/) (/) = (/)))
13	10, 12	mtbiri 715	. . 3 |- (A = (/) -> -. |^|_x e. A (/) = (/))
14	13	necon2ai 1603	. 2 |- (|^|_x e. A (/) = (/) -> A =/= (/))
15	4, 14	impbi 157	1 |- (A =/= (/) <-> |^|_x e. A (/) = (/))

Syntax hints:  -. wn 2   <-> wb 146   = wceq 953   e. wcel 955  {cab 1456   =/= wne 1577  A.wral 1637  Vcvv 1802  (/)c0 2270  |^|_ciin 2557

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-nul 2700

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-nul 2271  df-iin 2559

Copyright terms: Public domain