Theorem intexr 3986

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 3971	. . 3 |- -. _V e. _V
2		inteq 3691	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3702	. . . . 5 |- |^| (/) = _V
4	2, 3	syl6eq 2136	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2156	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 635	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2309	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1289   e. wcel 1438   =/= wne 2255   _Vcvv 2619   (/)c0 3286   |^|cint 3688

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-nul 3287  df-int 3689

This theorem is referenced by: (None)