Theorem ixxex 10095

Description: The set of intervals of extended reals exists. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)

Hypothesis

Ref	Expression
ixx.1	|- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )

Assertion

Ref	Expression
ixxex	|- O e. _V

Distinct variable groups:   x, y, z, R   x, S, y, z

Allowed substitution hints:   O( x, y, z)

Proof of Theorem ixxex

Step	Hyp	Ref	Expression
1		xrex 10052	. . . 4 |- RR* e. _V
2	1, 1	xpex 4834	. . 3 |- ( RR* X. RR* ) e. _V
3	1	pwex 4267	. . 3 |- ~P RR* e. _V
4	2, 3	xpex 4834	. 2 |- ( ( RR* X. RR* ) X. ~P RR* ) e. _V
5		ixx.1	. . . 4 |- O = ( x e. RR* , y e. RR* |-> { z e. RR* | ( x R z /\ z S y ) } )
6	5	ixxf 10094	. . 3 |- O : ( RR* X. RR* ) --> ~P RR*
7		fssxp 5491	. . 3 |- ( O : ( RR* X. RR* ) --> ~P RR* -> O C_ ( ( RR* X. RR* ) X. ~P RR* ) )
8	6, 7	ax-mp 5	. 2 |- O C_ ( ( RR* X. RR* ) X. ~P RR* )
9	4, 8	ssexi 4222	1 |- O e. _V

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   C_ wss 3197   ~Pcpw 3649   class class class wbr 4083   X. cxp 4717   -->wf 5314   e. cmpo 6003   RR*cxr 8180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8090  ax-resscn 8091

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-pnf 8183  df-mnf 8184  df-xr 8185

This theorem is referenced by:  iooex  10103