Theorem ixxex 9835

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 9792	. . . 4 |- RR* e. _V
2	1, 1	xpex 4719	. . 3 |- ( RR* X. RR* ) e. _V
3	1	pwex 4162	. . 3 |- ~P RR* e. _V
4	2, 3	xpex 4719	. 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 9834	. . 3 |- O : ( RR* X. RR* ) --> ~P RR*
7		fssxp 5355	. . 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 4120	1 |- O e. _V

Syntax hints:   /\ wa 103   = wceq 1343   e. wcel 2136   {crab 2448   _Vcvv 2726   C_ wss 3116   ~Pcpw 3559   class class class wbr 3982   X. cxp 4602   -->wf 5184   e. cmpo 5844   RR*cxr 7932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-cnex 7844  ax-resscn 7845

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937

This theorem is referenced by:  iooex  9843