Theorem intex 4523

Description: The intersection of a nonempty class exists. Exercise 5 of [TakeutiZaring] p. 44 and its converse. (Contributed by NM, 13-Aug-2002.)

Assertion

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

Proof of Theorem intex

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3714	. . 3 |- ( A =/= (/) <-> E. x x e. A )
2		intss1 4213	. . . . 5 |- ( x e. A -> |^| A C_ x )
3		vex 3025	. . . . . 6 |- x e. _V
4	3	ssex 4511	. . . . 5 |- ( |^| A C_ x -> |^| A e. _V )
5	2, 4	syl 17	. . . 4 |- ( x e. A -> |^| A e. _V )
6	5	exlimiv 1770	. . 3 |- ( E. x x e. A -> |^| A e. _V )
7	1, 6	sylbi 198	. 2 |- ( A =/= (/) -> |^| A e. _V )
8		vprc 4505	. . . 4 |- -. _V e. _V
9		inteq 4201	. . . . . 6 |- ( A = (/) -> |^| A = |^| (/) )
10		int0 4212	. . . . . 6 |- |^| (/) = _V
11	9, 10	syl6eq 2478	. . . . 5 |- ( A = (/) -> |^| A = _V )
12	11	eleq1d 2490	. . . 4 |- ( A = (/) -> ( |^| A e. _V <-> _V e. _V ) )
13	8, 12	mtbiri 304	. . 3 |- ( A = (/) -> -. |^| A e. _V )
14	13	necon2ai 2630	. 2 |- ( |^| A e. _V -> A =/= (/) )
15	7, 14	impbii 190	1 |- ( A =/= (/) <-> |^| A e. _V )

Syntax hints:   <-> wb 187   = wceq 1437   E.wex 1657   e. wcel 1872   =/= wne 2599   _Vcvv 3022   C_ wss 3379   (/)c0 3704   |^|cint 4198

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-v 3024  df-dif 3382  df-in 3386  df-ss 3393  df-nul 3705  df-int 4199

This theorem is referenced by:  intnex  4524  intexab  4525  iinexg  4527  onint0  6581  onintrab  6586  onmindif2  6597  fival  7879  elfi2  7881  elfir  7882  dffi2  7890  elfiun  7897  fifo  7899  tz9.1c  8166  tz9.12lem1  8210  tz9.12lem3  8212  rankf  8217  cardf2  8329  cardval3  8338  cardid2  8339  cardcf  8633  cflim2  8644  intwun  9111  wuncval  9118  inttsk  9150  intgru  9190  gruina  9194  dfrtrcl2  13069  mremre  15453  mrcval  15459  asplss  18496  aspsubrg  18498  toponmre  20051  subbascn  20212  insiga  28911  sigagenval  28914  sigagensiga  28915  dmsigagen  28918  dfon2lem8  30387  dfon2lem9  30388  igenval  32201  pclvalN  33367  elrfi  35448  ismrcd1  35452  mzpval  35486  dmmzp  35487  salgenval  38046  intsal  38053