Theorem iin0 4839

Description: An indexed intersection of the empty set, with a nonempty index set, is empty. (Contributed by NM, 20-Oct-2005.)

Assertion

Ref	Expression
iin0	|- ( A =/= (/) <-> |^|_ x e. A (/) = (/) )

Distinct variable group:   x, A

Proof of Theorem iin0

Step	Hyp	Ref	Expression
1		iinconst 4530	. 2 |- ( A =/= (/) -> |^|_ x e. A (/) = (/) )
2		0ex 4790	. . . . . 6 |- (/) e. _V
3	2	n0ii 3922	. . . . 5 |- -. _V = (/)
4		0iin 4578	. . . . . 6 |- |^|_ x e. (/) (/) = _V
5	4	eqeq1i 2627	. . . . 5 |- ( |^|_ x e. (/) (/) = (/) <-> _V = (/) )
6	3, 5	mtbir 313	. . . 4 |- -. |^|_ x e. (/) (/) = (/)
7		iineq1 4535	. . . . 5 |- ( A = (/) -> |^|_ x e. A (/) = |^|_ x e. (/) (/) )
8	7	eqeq1d 2624	. . . 4 |- ( A = (/) -> ( |^|_ x e. A (/) = (/) <-> |^|_ x e. (/) (/) = (/) ) )
9	6, 8	mtbiri 317	. . 3 |- ( A = (/) -> -. |^|_ x e. A (/) = (/) )
10	9	necon2ai 2823	. 2 |- ( |^|_ x e. A (/) = (/) -> A =/= (/) )
11	1, 10	impbii 199	1 |- ( A =/= (/) <-> |^|_ x e. A (/) = (/) )

Syntax hints:   <-> wb 196   = wceq 1483   =/= wne 2794   _Vcvv 3200   (/)c0 3915   |^|_ciin 4521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-nul 3916  df-iin 4523

This theorem is referenced by: (None)