Theorem intexr 4213

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 4195	. . 3 |- -. _V e. _V
2		inteq 3905	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3916	. . . . 5 |- |^| (/) = _V
4	2, 3	eqtrdi 2258	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2278	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 679	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2434	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1375   e. wcel 2180   =/= wne 2380   _Vcvv 2779   (/)c0 3471   |^|cint 3902

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-v 2781  df-dif 3179  df-nul 3472  df-int 3903

This theorem is referenced by:  fival  7105