Theorem iin0 3477

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 2962	. . . 4 |- (A =/= (/) -> (y e. (/) <-> A.x e. A y e. (/)))
2	1	abbi2dv 2009	. . 3 |- (A =/= (/) -> (/) = {y | A.x e. A y e. (/)})
3		df-iin 3258	. . 3 |- |^|_x e. A (/) = {y | A.x e. A y e. (/)}
4	2, 3	syl6reqr 1947	. 2 |- (A =/= (/) -> |^|_x e. A (/) = (/))
5		0ex 3446	. . . . . 6 |- (/) e. _V
6		n0i 2880	. . . . . 6 |- ((/) e. _V -> -. _V = (/))
7	5, 6	ax-mp 7	. . . . 5 |- -. _V = (/)
8		0iin 3313	. . . . . 6 |- |^|_x e. (/) (/) = _V
9	8	eqeq1i 1891	. . . . 5 |- (|^|_x e. (/) (/) = (/) <-> _V = (/))
10	7, 9	mtbir 209	. . . 4 |- -. |^|_x e. (/) (/) = (/)
11		iineq1 3270	. . . . 5 |- (A = (/) -> |^|_x e. A (/) = |^|_x e. (/) (/))
12	11	eqeq1d 1892	. . . 4 |- (A = (/) -> (|^|_x e. A (/) = (/) <-> |^|_x e. (/) (/) = (/)))
13	10, 12	mtbiri 785	. . 3 |- (A = (/) -> -. |^|_x e. A (/) = (/))
14	13	necon2ai 2051	. 2 |- (|^|_x e. A (/) = (/) -> A =/= (/))
15	4, 14	impbii 174	1 |- (A =/= (/) <-> |^|_x e. A (/) = (/))

Syntax hints:  -. wn 2   <-> wb 163   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017  A.wral 2105  _Vcvv 2292  (/)c0 2875  |^|_ciin 3256

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-nul 3445

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-nul 2876  df-iin 3258

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