Theorem intnex 4821

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 4820	. . . 4 |- ( A =/= (/) <-> |^| A e. _V )
2	1	necon1bbii 2843	. . 3 |- ( -. |^| A e. _V <-> A = (/) )
3		inteq 4478	. . . 4 |- ( A = (/) -> |^| A = |^| (/) )
4		int0 4490	. . . 4 |- |^| (/) = _V
5	3, 4	syl6eq 2672	. . 3 |- ( A = (/) -> |^| A = _V )
6	2, 5	sylbi 207	. 2 |- ( -. |^| A e. _V -> |^| A = _V )
7		vprc 4796	. . 3 |- -. _V e. _V
8		eleq1 2689	. . 3 |- ( |^| A = _V -> ( |^| A e. _V <-> _V e. _V ) )
9	7, 8	mtbiri 317	. 2 |- ( |^| A = _V -> -. |^| A e. _V )
10	6, 9	impbii 199	1 |- ( -. |^| A e. _V <-> |^| A = _V )

Syntax hints:   -. wn 3   <-> wb 196   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   |^|cint 4475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-int 4476

This theorem is referenced by:  intabs  4825  relintabex  37887