Theorem intnex 4613

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 4612	. . . 4 |- ( A =/= (/) <-> |^| A e. _V )
2	1	necon1bbii 2721	. . 3 |- ( -. |^| A e. _V <-> A = (/) )
3		inteq 4291	. . . 4 |- ( A = (/) -> |^| A = |^| (/) )
4		int0 4302	. . . 4 |- |^| (/) = _V
5	3, 4	syl6eq 2514	. . 3 |- ( A = (/) -> |^| A = _V )
6	2, 5	sylbi 195	. 2 |- ( -. |^| A e. _V -> |^| A = _V )
7		vprc 4594	. . 3 |- -. _V e. _V
8		eleq1 2529	. . 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 1395   e. wcel 1819   _Vcvv 3109   (/)c0 3793   |^|cint 4288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-v 3111  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794  df-int 4289

This theorem is referenced by:  intabs  4617