Theorem iin0r 4155

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 4116	. . . . 5 |- (/) e. _V
2		n0i 3420	. . . . 5 |- ( (/) e. _V -> -. _V = (/) )
3	1, 2	ax-mp 5	. . . 4 |- -. _V = (/)
4		0iin 3931	. . . . 5 |- |^|_ x e. (/) (/) = _V
5	4	eqeq1i 2178	. . . 4 |- ( |^|_ x e. (/) (/) = (/) <-> _V = (/) )
6	3, 5	mtbir 666	. . 3 |- -. |^|_ x e. (/) (/) = (/)
7		iineq1 3887	. . . 4 |- ( A = (/) -> |^|_ x e. A (/) = |^|_ x e. (/) (/) )
8	7	eqeq1d 2179	. . 3 |- ( A = (/) -> ( |^|_ x e. A (/) = (/) <-> |^|_ x e. (/) (/) = (/) ) )
9	6, 8	mtbiri 670	. 2 |- ( A = (/) -> -. |^|_ x e. A (/) = (/) )
10	9	necon2ai 2394	1 |- ( |^|_ x e. A (/) = (/) -> A =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1348   e. wcel 2141   =/= wne 2340   _Vcvv 2730   (/)c0 3414   |^|_ciin 3874

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-nul 3415  df-iin 3876

This theorem is referenced by: (None)