Theorem intnex 4558

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 4557	. . . 4 |- ( A =/= (/) <-> |^| A e. _V )
2	1	necon1bbii 2692	. . 3 |- ( -. |^| A e. _V <-> A = (/) )
3		inteq 4229	. . . 4 |- ( A = (/) -> |^| A = |^| (/) )
4		int0 4240	. . . 4 |- |^| (/) = _V
5	3, 4	syl6eq 2521	. . 3 |- ( A = (/) -> |^| A = _V )
6	2, 5	sylbi 200	. 2 |- ( -. |^| A e. _V -> |^| A = _V )
7		vprc 4534	. . 3 |- -. _V e. _V
8		eleq1 2537	. . 3 |- ( |^| A = _V -> ( |^| A e. _V <-> _V e. _V ) )
9	7, 8	mtbiri 310	. 2 |- ( |^| A = _V -> -. |^| A e. _V )
10	6, 9	impbii 192	1 |- ( -. |^| A e. _V <-> |^| A = _V )

Syntax hints:   -. wn 3   <-> wb 189   = wceq 1452   e. wcel 1904   _Vcvv 3031   (/)c0 3722   |^|cint 4226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-int 4227

This theorem is referenced by:  intabs  4562  relintabex  36258