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Theorem ssxr 9101
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 3781 . . . . . . 7  |-  {  +oo , 
-oo }  =  ( {  +oo }  u.  {  -oo } )
21ineq2i 3499 . . . . . 6  |-  ( A  i^i  {  +oo ,  -oo } )  =  ( A  i^i  ( { 
+oo }  u.  {  -oo } ) )
3 indi 3547 . . . . . 6  |-  ( A  i^i  ( {  +oo }  u.  {  -oo }
) )  =  ( ( A  i^i  {  +oo } )  u.  ( A  i^i  {  -oo }
) )
42, 3eqtri 2424 . . . . 5  |-  ( A  i^i  {  +oo ,  -oo } )  =  ( ( A  i^i  {  +oo } )  u.  ( A  i^i  {  -oo }
) )
5 disjsn 3828 . . . . . . . 8  |-  ( ( A  i^i  {  +oo } )  =  (/)  <->  -.  +oo  e.  A )
6 disjsn 3828 . . . . . . . 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 3623 . . . . . 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 2448 . . . 4  |-  ( -.  (  +oo  e.  A  \/  -oo  e.  A )  ->  ( A  i^i  { 
+oo ,  -oo } )  =  (/) )
13 reldisj 3631 . . . . 5  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( ( A  i^i  {  +oo ,  -oo } )  =  (/)  <->  A  C_  ( ( RR  u.  { 
+oo ,  -oo } ) 
\  {  +oo ,  -oo } ) ) )
14 renfdisj 9094 . . . . . . . 8  |-  ( RR 
i^i  {  +oo ,  -oo } )  =  (/)
15 disj3 3632 . . . . . . . 8  |-  ( ( RR  i^i  {  +oo , 
-oo } )  =  (/)  <->  RR  =  ( RR  \  {  +oo ,  -oo }
) )
1614, 15mpbi 200 . . . . . . 7  |-  RR  =  ( RR  \  {  +oo , 
-oo } )
17 difun2 3667 . . . . . . 7  |-  ( ( RR  u.  {  +oo , 
-oo } )  \  {  +oo ,  -oo } )  =  ( RR  \  {  +oo ,  -oo }
)
1816, 17eqtr4i 2427 . . . . . 6  |-  RR  =  ( ( RR  u.  { 
+oo ,  -oo } ) 
\  {  +oo ,  -oo } )
1918sseq2i 3333 . . . . 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 9080 . . 3  |-  RR*  =  ( RR  u.  {  +oo , 
-oo } )
2423sseq2i 3333 . 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 1721    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   {cpr 3775   RRcr 8945    +oocpnf 9073    -oocmnf 9074   RR*cxr 9075
This theorem is referenced by:  xrsupss  10843  xrinfmss  10844
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080
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