Theorem intexr 4007

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 3992	. . 3 |- -. _V e. _V
2		inteq 3713	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3724	. . . . 5 |- |^| (/) = _V
4	2, 3	syl6eq 2143	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2163	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 638	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2316	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1296   e. wcel 1445   =/= wne 2262   _Vcvv 2633   (/)c0 3302   |^|cint 3710

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-v 2635  df-dif 3015  df-nul 3303  df-int 3711

This theorem is referenced by: (None)