Theorem intexr 4198

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 4180	. . 3 |- -. _V e. _V
2		inteq 3890	. . . . 5 |- ( A = (/) -> |^| A = |^| (/) )
3		int0 3901	. . . . 5 |- |^| (/) = _V
4	2, 3	eqtrdi 2255	. . . 4 |- ( A = (/) -> |^| A = _V )
5	4	eleq1d 2275	. . 3 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
6	1, 5	mtbiri 677	. 2 |- ( A = (/) -> -. |^| A e. _V )
7	6	necon2ai 2431	1 |- ( |^| A e. _V -> A =/= (/) )

Syntax hints:   -> wi 4   = wceq 1373   e. wcel 2177   =/= wne 2377   _Vcvv 2773   (/)c0 3461   |^|cint 3887

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-v 2775  df-dif 3169  df-nul 3462  df-int 3888

This theorem is referenced by:  fival  7079