Theorem iin0r 4171

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 4132	. . . . 5 |- (/) e. _V
2		n0i 3430	. . . . 5 |- ( (/) e. _V -> -. _V = (/) )
3	1, 2	ax-mp 5	. . . 4 |- -. _V = (/)
4		0iin 3947	. . . . 5 |- |^|_ x e. (/) (/) = _V
5	4	eqeq1i 2185	. . . 4 |- ( |^|_ x e. (/) (/) = (/) <-> _V = (/) )
6	3, 5	mtbir 671	. . 3 |- -. |^|_ x e. (/) (/) = (/)
7		iineq1 3902	. . . 4 |- ( A = (/) -> |^|_ x e. A (/) = |^|_ x e. (/) (/) )
8	7	eqeq1d 2186	. . 3 |- ( A = (/) -> ( |^|_ x e. A (/) = (/) <-> |^|_ x e. (/) (/) = (/) ) )
9	6, 8	mtbiri 675	. 2 |- ( A = (/) -> -. |^|_ x e. A (/) = (/) )
10	9	necon2ai 2401	1 |- ( |^|_ x e. A (/) = (/) -> A =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   = wceq 1353   e. wcel 2148   =/= wne 2347   _Vcvv 2739   (/)c0 3424   |^|_ciin 3889

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-nul 4131

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2741  df-dif 3133  df-nul 3425  df-iin 3891

This theorem is referenced by: (None)