Theorem intexr 4136

Description: If the intersection of a class exists, the class is nonempty. (Contributed by Jim Kingdon, 27-Aug-2018.)

Assertion

Ref	Expression
intexr	|- ( |^| A e. _V -> A =/= (/) )

Proof of Theorem intexr

Step	Hyp	Ref	Expression
1		vprc 4121	. . 3 |- -. _V e. _V
2		inteq 3834	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3845	. . . . 5 |- |^| (/) = _V
4	2, 3	eqtrdi 2219	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2239	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 670	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2394	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   =/= wne 2340   _Vcvv 2730   (/)c0 3414   |^|cint 3831

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-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  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-int 3832

This theorem is referenced by:  fival  6947