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Theorem ssxr 9078
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
StepHypRef Expression
1 df-pr 3764 . . . . . . 7  |-  {  +oo , 
-oo }  =  ( {  +oo }  u.  {  -oo } )
21ineq2i 3482 . . . . . 6  |-  ( A  i^i  {  +oo ,  -oo } )  =  ( A  i^i  ( { 
+oo }  u.  {  -oo } ) )
3 indi 3530 . . . . . 6  |-  ( A  i^i  ( {  +oo }  u.  {  -oo }
) )  =  ( ( A  i^i  {  +oo } )  u.  ( A  i^i  {  -oo }
) )
42, 3eqtri 2407 . . . . 5  |-  ( A  i^i  {  +oo ,  -oo } )  =  ( ( A  i^i  {  +oo } )  u.  ( A  i^i  {  -oo }
) )
5 disjsn 3811 . . . . . . . 8  |-  ( ( A  i^i  {  +oo } )  =  (/)  <->  -.  +oo  e.  A )
6 disjsn 3811 . . . . . . . 8  |-  ( ( A  i^i  {  -oo } )  =  (/)  <->  -.  -oo  e.  A )
75, 6anbi12i 679 . . . . . . 7  |-  ( ( ( A  i^i  {  +oo } )  =  (/)  /\  ( A  i^i  {  -oo } )  =  (/) ) 
<->  ( -.  +oo  e.  A  /\  -.  -oo  e.  A ) )
87biimpri 198 . . . . . 6  |-  ( ( -.  +oo  e.  A  /\  -.  -oo  e.  A
)  ->  ( ( A  i^i  {  +oo }
)  =  (/)  /\  ( A  i^i  {  -oo }
)  =  (/) ) )
9 pm4.56 482 . . . . . 6  |-  ( ( -.  +oo  e.  A  /\  -.  -oo  e.  A
)  <->  -.  (  +oo  e.  A  \/  -oo  e.  A ) )
10 un00 3606 . . . . . 6  |-  ( ( ( A  i^i  {  +oo } )  =  (/)  /\  ( A  i^i  {  -oo } )  =  (/) ) 
<->  ( ( A  i^i  { 
+oo } )  u.  ( A  i^i  {  -oo }
) )  =  (/) )
118, 9, 103imtr3i 257 . . . . 5  |-  ( -.  (  +oo  e.  A  \/  -oo  e.  A )  ->  ( ( A  i^i  {  +oo }
)  u.  ( A  i^i  {  -oo }
) )  =  (/) )
124, 11syl5eq 2431 . . . 4  |-  ( -.  (  +oo  e.  A  \/  -oo  e.  A )  ->  ( A  i^i  { 
+oo ,  -oo } )  =  (/) )
13 reldisj 3614 . . . . 5  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( ( A  i^i  {  +oo ,  -oo } )  =  (/)  <->  A  C_  ( ( RR  u.  { 
+oo ,  -oo } ) 
\  {  +oo ,  -oo } ) ) )
14 renfdisj 9071 . . . . . . . 8  |-  ( RR 
i^i  {  +oo ,  -oo } )  =  (/)
15 disj3 3615 . . . . . . . 8  |-  ( ( RR  i^i  {  +oo , 
-oo } )  =  (/)  <->  RR  =  ( RR  \  {  +oo ,  -oo }
) )
1614, 15mpbi 200 . . . . . . 7  |-  RR  =  ( RR  \  {  +oo , 
-oo } )
17 difun2 3650 . . . . . . 7  |-  ( ( RR  u.  {  +oo , 
-oo } )  \  {  +oo ,  -oo } )  =  ( RR  \  {  +oo ,  -oo }
)
1816, 17eqtr4i 2410 . . . . . 6  |-  RR  =  ( ( RR  u.  { 
+oo ,  -oo } ) 
\  {  +oo ,  -oo } )
1918sseq2i 3316 . . . . 5  |-  ( A 
C_  RR  <->  A  C_  ( ( RR  u.  {  +oo , 
-oo } )  \  {  +oo ,  -oo } ) )
2013, 19syl6bbr 255 . . . 4  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( ( A  i^i  {  +oo ,  -oo } )  =  (/)  <->  A  C_  RR ) )
2112, 20syl5ib 211 . . 3  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( -.  (  +oo  e.  A  \/  -oo  e.  A )  ->  A  C_  RR ) )
2221orrd 368 . 2  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( (  +oo  e.  A  \/  -oo  e.  A )  \/  A  C_  RR ) )
23 df-xr 9057 . . 3  |-  RR*  =  ( RR  u.  {  +oo , 
-oo } )
2423sseq2i 3316 . 2  |-  ( A 
C_  RR*  <->  A  C_  ( RR  u.  {  +oo ,  -oo } ) )
25 3orrot 942 . . 3  |-  ( ( A  C_  RR  \/  +oo 
e.  A  \/  -oo  e.  A )  <->  (  +oo  e.  A  \/  -oo  e.  A  \/  A  C_  RR ) )
26 df-3or 937 . . 3  |-  ( ( 
+oo  e.  A  \/  -oo 
e.  A  \/  A  C_  RR )  <->  ( (  +oo  e.  A  \/  -oo  e.  A )  \/  A  C_  RR ) )
2725, 26bitri 241 . 2  |-  ( ( A  C_  RR  \/  +oo 
e.  A  \/  -oo  e.  A )  <->  ( (  +oo  e.  A  \/  -oo  e.  A )  \/  A  C_  RR ) )
2822, 24, 273imtr4i 258 1  |-  ( A 
C_  RR*  ->  ( A  C_  RR  \/  +oo  e.  A  \/  -oo  e.  A
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1717    \ cdif 3260    u. cun 3261    i^i cin 3262    C_ wss 3263   (/)c0 3571   {csn 3757   {cpr 3758   RRcr 8922    +oocpnf 9050    -oocmnf 9051   RR*cxr 9052
This theorem is referenced by:  xrsupss  10819  xrinfmss  10820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641  ax-resscn 8980
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-er 6841  df-en 7046  df-dom 7047  df-sdom 7048  df-pnf 9055  df-mnf 9056  df-xr 9057
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