Theorem iin0r 4265

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 4221	. . . . 5 |- (/) e. _V
2		n0i 3502	. . . . 5 |- ( (/) e. _V -> -. _V = (/) )
3	1, 2	ax-mp 5	. . . 4 |- -. _V = (/)
4		0iin 4034	. . . . 5 |- |^|_ x e. (/) (/) = _V
5	4	eqeq1i 2239	. . . 4 |- ( |^|_ x e. (/) (/) = (/) <-> _V = (/) )
6	3, 5	mtbir 678	. . 3 |- -. |^|_ x e. (/) (/) = (/)
7		iineq1 3989	. . . 4 |- ( A = (/) -> |^|_ x e. A (/) = |^|_ x e. (/) (/) )
8	7	eqeq1d 2240	. . 3 |- ( A = (/) -> ( |^|_ x e. A (/) = (/) <-> |^|_ x e. (/) (/) = (/) ) )
9	6, 8	mtbiri 682	. 2 |- ( A = (/) -> -. |^|_ x e. A (/) = (/) )
10	9	necon2ai 2457	1 |- ( |^|_ x e. A (/) = (/) -> A =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1398   e. wcel 2202   =/= wne 2403   _Vcvv 2803   (/)c0 3496   |^|_ciin 3976

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213  ax-nul 4220

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-v 2805  df-dif 3203  df-nul 3497  df-iin 3978

This theorem is referenced by: (None)