Theorem intexr 4264

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 4244	. . 3 |- -. _V e. _V
2		inteq 3954	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3965	. . . . 5 |- |^| (/) = _V
4	2, 3	eqtrdi 2283	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2303	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 682	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2468	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   =/= wne 2414   _Vcvv 2815   (/)c0 3510   |^|cint 3951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-v 2817  df-dif 3215  df-nul 3511  df-int 3952

This theorem is referenced by:  fival  7259