Theorem intexr 4241

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 4222	. . 3 |- -. _V e. _V
2		inteq 3932	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3943	. . . . 5 |- |^| (/) = _V
4	2, 3	eqtrdi 2279	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2299	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 681	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2455	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2201   =/= wne 2401   _Vcvv 2801   (/)c0 3493   |^|cint 3929

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-v 2803  df-dif 3201  df-nul 3494  df-int 3930

This theorem is referenced by:  fival  7174