Theorem iin0 2736

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 2345	. . . 4 |- (A =/= (/) -> (y e. (/) <-> A.x e. A y e. (/)))
2	1	abbi2dv 1576	. . 3 |- (A =/= (/) -> (/) = {y | A.x e. A y e. (/)})
3		df-iin 2565	. . 3 |- |^|_x e. A (/) = {y | A.x e. A y e. (/)}
4	2, 3	syl6reqr 1524	. 2 |- (A =/= (/) -> |^|_x e. A (/) = (/))
5		0ex 2707	. . . . . 6 |- (/) e. V
6		n0i 2282	. . . . . 6 |- ((/) e. V -> -. V = (/))
7	5, 6	ax-mp 7	. . . . 5 |- -. V = (/)
8		0iin 2602	. . . . . 6 |- |^|_x e. (/) (/) = V
9	8	eqeq1i 1480	. . . . 5 |- (|^|_x e. (/) (/) = (/) <-> V = (/))
10	7, 9	mtbir 192	. . . 4 |- -. |^|_x e. (/) (/) = (/)
11		iineq1 2572	. . . . 5 |- (A = (/) -> |^|_x e. A (/) = |^|_x e. (/) (/))
12	11	eqeq1d 1481	. . . 4 |- (A = (/) -> (|^|_x e. A (/) = (/) <-> |^|_x e. (/) (/) = (/)))
13	10, 12	mtbiri 716	. . 3 |- (A = (/) -> -. |^|_x e. A (/) = (/))
14	13	necon2ai 1609	. 2 |- (|^|_x e. A (/) = (/) -> A =/= (/))
15	4, 14	impbi 157	1 |- (A =/= (/) <-> |^|_x e. A (/) = (/))

Syntax hints:  -. wn 2   <-> wb 146   = wceq 955   e. wcel 957  {cab 1462   =/= wne 1583  A.wral 1643  Vcvv 1808  (/)c0 2277  |^|_ciin 2563

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-nul 2706

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-v 1809  df-dif 2046  df-nul 2278  df-iin 2565

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