Theorem iin0r 4093

Description: If an indexed intersection of the empty set is empty, the index set is nonempty. (Contributed by Jim Kingdon, 29-Aug-2018.)

Assertion

Ref	Expression
iin0r	|- ( |^|_ x e. A (/) = (/) -> A =/= (/) )

Distinct variable group:   x, A

Proof of Theorem iin0r

Step	Hyp	Ref	Expression
1		0ex 4055	. . . . 5 |- (/) e. _V
2		n0i 3368	. . . . 5 |- ( (/) e. _V -> -. _V = (/) )
3	1, 2	ax-mp 5	. . . 4 |- -. _V = (/)
4		0iin 3871	. . . . 5 |- |^|_ x e. (/) (/) = _V
5	4	eqeq1i 2147	. . . 4 |- ( |^|_ x e. (/) (/) = (/) <-> _V = (/) )
6	3, 5	mtbir 660	. . 3 |- -. |^|_ x e. (/) (/) = (/)
7		iineq1 3827	. . . 4 |- ( A = (/) -> |^|_ x e. A (/) = |^|_ x e. (/) (/) )
8	7	eqeq1d 2148	. . 3 |- ( A = (/) -> ( |^|_ x e. A (/) = (/) <-> |^|_ x e. (/) (/) = (/) ) )
9	6, 8	mtbiri 664	. 2 |- ( A = (/) -> -. |^|_ x e. A (/) = (/) )
10	9	necon2ai 2362	1 |- ( |^|_ x e. A (/) = (/) -> A =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1331   e. wcel 1480   =/= wne 2308   _Vcvv 2686   (/)c0 3363   |^|_ciin 3814

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-v 2688  df-dif 3073  df-nul 3364  df-iin 3816

This theorem is referenced by: (None)