Theorem iin0r 3922

Description: If an indexed intersection of the empty set is empty, the index set is non-empty. (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 3884	. . . . 5 |- (/) e. _V
2		n0i 3229	. . . . 5 |- ( (/) e. _V -> -. _V = (/) )
3	1, 2	ax-mp 7	. . . 4 |- -. _V = (/)
4		0iin 3715	. . . . 5 |- |^|_ x e. (/) (/) = _V
5	4	eqeq1i 2047	. . . 4 |- ( |^|_ x e. (/) (/) = (/) <-> _V = (/) )
6	3, 5	mtbir 596	. . 3 |- -. |^|_ x e. (/) (/) = (/)
7		iineq1 3671	. . . 4 |- ( A = (/) -> |^|_ x e. A (/) = |^|_ x e. (/) (/) )
8	7	eqeq1d 2048	. . 3 |- ( A = (/) -> ( |^|_ x e. A (/) = (/) <-> |^|_ x e. (/) (/) = (/) ) )
9	6, 8	mtbiri 600	. 2 |- ( A = (/) -> -. |^|_ x e. A (/) = (/) )
10	9	necon2ai 2259	1 |- ( |^|_ x e. A (/) = (/) -> A =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1243   e. wcel 1393   =/= wne 2204   _Vcvv 2557   (/)c0 3224   |^|_ciin 3658

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-nul 3225  df-iin 3660

This theorem is referenced by: (None)