Theorem ssxr 10107

Description: The three (non-exclusive) possibilities implied by a subset of extended reals. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)

Assertion

Ref	Expression
ssxr	|- ( A C_ RR* -> ( A C_ RR \/ +oo e. A \/ -oo e. A ) )

Proof of Theorem ssxr

Step	Hyp	Ref	Expression
1		df-pr 4180	. . . . . . 7 |- { +oo , -oo } = ( { +oo } u. { -oo } )
2	1	ineq2i 3811	. . . . . 6 |- ( A i^i { +oo , -oo } ) = ( A i^i ( { +oo } u. { -oo } ) )
3		indi 3873	. . . . . 6 |- ( A i^i ( { +oo } u. { -oo } ) ) = ( ( A i^i { +oo } ) u. ( A i^i { -oo } ) )
4	2, 3	eqtri 2644	. . . . 5 |- ( A i^i { +oo , -oo } ) = ( ( A i^i { +oo } ) u. ( A i^i { -oo } ) )
5		disjsn 4246	. . . . . . . 8 |- ( ( A i^i { +oo } ) = (/) <-> -. +oo e. A )
6		disjsn 4246	. . . . . . . 8 |- ( ( A i^i { -oo } ) = (/) <-> -. -oo e. A )
7	5, 6	anbi12i 733	. . . . . . 7 |- ( ( ( A i^i { +oo } ) = (/) /\ ( A i^i { -oo } ) = (/) ) <-> ( -. +oo e. A /\ -. -oo e. A ) )
8	7	biimpri 218	. . . . . 6 |- ( ( -. +oo e. A /\ -. -oo e. A ) -> ( ( A i^i { +oo } ) = (/) /\ ( A i^i { -oo } ) = (/) ) )
9		pm4.56 516	. . . . . 6 |- ( ( -. +oo e. A /\ -. -oo e. A ) <-> -. ( +oo e. A \/ -oo e. A ) )
10		un00 4011	. . . . . 6 |- ( ( ( A i^i { +oo } ) = (/) /\ ( A i^i { -oo } ) = (/) ) <-> ( ( A i^i { +oo } ) u. ( A i^i { -oo } ) ) = (/) )
11	8, 9, 10	3imtr3i 280	. . . . 5 |- ( -. ( +oo e. A \/ -oo e. A ) -> ( ( A i^i { +oo } ) u. ( A i^i { -oo } ) ) = (/) )
12	4, 11	syl5eq 2668	. . . 4 |- ( -. ( +oo e. A \/ -oo e. A ) -> ( A i^i { +oo , -oo } ) = (/) )
13		reldisj 4020	. . . . 5 |- ( A C_ ( RR u. { +oo , -oo } ) -> ( ( A i^i { +oo , -oo } ) = (/) <-> A C_ ( ( RR u. { +oo , -oo } ) \ { +oo , -oo } ) ) )
14		renfdisj 10098	. . . . . . . 8 |- ( RR i^i { +oo , -oo } ) = (/)
15		disj3 4021	. . . . . . . 8 |- ( ( RR i^i { +oo , -oo } ) = (/) <-> RR = ( RR \ { +oo , -oo } ) )
16	14, 15	mpbi 220	. . . . . . 7 |- RR = ( RR \ { +oo , -oo } )
17		difun2 4048	. . . . . . 7 |- ( ( RR u. { +oo , -oo } ) \ { +oo , -oo } ) = ( RR \ { +oo , -oo } )
18	16, 17	eqtr4i 2647	. . . . . 6 |- RR = ( ( RR u. { +oo , -oo } ) \ { +oo , -oo } )
19	18	sseq2i 3630	. . . . 5 |- ( A C_ RR <-> A C_ ( ( RR u. { +oo , -oo } ) \ { +oo , -oo } ) )
20	13, 19	syl6bbr 278	. . . 4 |- ( A C_ ( RR u. { +oo , -oo } ) -> ( ( A i^i { +oo , -oo } ) = (/) <-> A C_ RR ) )
21	12, 20	syl5ib 234	. . 3 |- ( A C_ ( RR u. { +oo , -oo } ) -> ( -. ( +oo e. A \/ -oo e. A ) -> A C_ RR ) )
22	21	orrd 393	. 2 |- ( A C_ ( RR u. { +oo , -oo } ) -> ( ( +oo e. A \/ -oo e. A ) \/ A C_ RR ) )
23		df-xr 10078	. . 3 |- RR* = ( RR u. { +oo , -oo } )
24	23	sseq2i 3630	. 2 |- ( A C_ RR* <-> A C_ ( RR u. { +oo , -oo } ) )
25		3orrot 1044	. . 3 |- ( ( A C_ RR \/ +oo e. A \/ -oo e. A ) <-> ( +oo e. A \/ -oo e. A \/ A C_ RR ) )
26		df-3or 1038	. . 3 |- ( ( +oo e. A \/ -oo e. A \/ A C_ RR ) <-> ( ( +oo e. A \/ -oo e. A ) \/ A C_ RR ) )
27	25, 26	bitri 264	. 2 |- ( ( A C_ RR \/ +oo e. A \/ -oo e. A ) <-> ( ( +oo e. A \/ -oo e. A ) \/ A C_ RR ) )
28	22, 24, 27	3imtr4i 281	1 |- ( A C_ RR* -> ( A C_ RR \/ +oo e. A \/ -oo e. A ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   {cpr 4179   RRcr 9935   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073

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  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078

This theorem is referenced by:  xrsupss  12139  xrinfmss  12140  xrssre  39564