Theorem intnex 4604

Description: If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)

Assertion

Ref	Expression
intnex	|- ( -. |^| A e. _V <-> |^| A = _V )

Proof of Theorem intnex

Step	Hyp	Ref	Expression
1		intex 4603	. . . 4 |- ( A =/= (/) <-> |^| A e. _V )
2	1	necon1bbii 2731	. . 3 |- ( -. |^| A e. _V <-> A = (/) )
3		inteq 4285	. . . 4 |- ( A = (/) -> |^| A = |^| (/) )
4		int0 4296	. . . 4 |- |^| (/) = _V
5	3, 4	syl6eq 2524	. . 3 |- ( A = (/) -> |^| A = _V )
6	2, 5	sylbi 195	. 2 |- ( -. |^| A e. _V -> |^| A = _V )
7		vprc 4585	. . 3 |- -. _V e. _V
8		eleq1 2539	. . 3 |- ( |^| A = _V -> ( |^| A e. _V <-> _V e. _V ) )
9	7, 8	mtbiri 303	. 2 |- ( |^| A = _V -> -. |^| A e. _V )
10	6, 9	impbii 188	1 |- ( -. |^| A e. _V <-> |^| A = _V )

Syntax hints:   -. wn 3   <-> wb 184   = wceq 1379   e. wcel 1767   _Vcvv 3113   (/)c0 3785   |^|cint 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-v 3115  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786  df-int 4283

This theorem is referenced by:  intabs  4608