Theorem intexr 4075

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 4060	. . 3 |- -. _V e. _V
2		inteq 3774	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3785	. . . . 5 |- |^| (/) = _V
4	2, 3	syl6eq 2188	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2208	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 664	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2362	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1331   e. wcel 1480   =/= wne 2308   _Vcvv 2686   (/)c0 3363   |^|cint 3771

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-v 2688  df-dif 3073  df-nul 3364  df-int 3772

This theorem is referenced by:  fival  6858