Theorem ixxex 9781

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 9738	. . . 4 |- RR* e. _V
2	1, 1	xpex 4694	. . 3 |- ( RR* X. RR* ) e. _V
3	1	pwex 4139	. . 3 |- ~P RR* e. _V
4	2, 3	xpex 4694	. 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 9780	. . 3 |- O : ( RR* X. RR* ) --> ~P RR*
7		fssxp 5330	. . 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 4098	1 |- O e. _V

Syntax hints:   /\ wa 103   = wceq 1332   e. wcel 2125   {crab 2436   _Vcvv 2709   C_ wss 3098   ~Pcpw 3539   class class class wbr 3961   X. cxp 4577   -->wf 5159   e. cmpo 5816   RR*cxr 7890

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-cnex 7802  ax-resscn 7803

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-fv 5171  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-pnf 7893  df-mnf 7894  df-xr 7895

This theorem is referenced by:  iooex  9789