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Theorem ssxr 8908
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 3660 . . . . . . 7  |-  {  +oo , 
-oo }  =  ( {  +oo }  u.  {  -oo } )
21ineq2i 3380 . . . . . 6  |-  ( A  i^i  {  +oo ,  -oo } )  =  ( A  i^i  ( { 
+oo }  u.  {  -oo } ) )
3 indi 3428 . . . . . 6  |-  ( A  i^i  ( {  +oo }  u.  {  -oo }
) )  =  ( ( A  i^i  {  +oo } )  u.  ( A  i^i  {  -oo }
) )
42, 3eqtri 2316 . . . . 5  |-  ( A  i^i  {  +oo ,  -oo } )  =  ( ( A  i^i  {  +oo } )  u.  ( A  i^i  {  -oo }
) )
5 disjsn 3706 . . . . . . . 8  |-  ( ( A  i^i  {  +oo } )  =  (/)  <->  -.  +oo  e.  A )
6 disjsn 3706 . . . . . . . 8  |-  ( ( A  i^i  {  -oo } )  =  (/)  <->  -.  -oo  e.  A )
75, 6anbi12i 678 . . . . . . 7  |-  ( ( ( A  i^i  {  +oo } )  =  (/)  /\  ( A  i^i  {  -oo } )  =  (/) ) 
<->  ( -.  +oo  e.  A  /\  -.  -oo  e.  A ) )
87biimpri 197 . . . . . 6  |-  ( ( -.  +oo  e.  A  /\  -.  -oo  e.  A
)  ->  ( ( A  i^i  {  +oo }
)  =  (/)  /\  ( A  i^i  {  -oo }
)  =  (/) ) )
9 pm4.56 481 . . . . . 6  |-  ( ( -.  +oo  e.  A  /\  -.  -oo  e.  A
)  <->  -.  (  +oo  e.  A  \/  -oo  e.  A ) )
10 un00 3503 . . . . . 6  |-  ( ( ( A  i^i  {  +oo } )  =  (/)  /\  ( A  i^i  {  -oo } )  =  (/) ) 
<->  ( ( A  i^i  { 
+oo } )  u.  ( A  i^i  {  -oo }
) )  =  (/) )
118, 9, 103imtr3i 256 . . . . 5  |-  ( -.  (  +oo  e.  A  \/  -oo  e.  A )  ->  ( ( A  i^i  {  +oo }
)  u.  ( A  i^i  {  -oo }
) )  =  (/) )
124, 11syl5eq 2340 . . . 4  |-  ( -.  (  +oo  e.  A  \/  -oo  e.  A )  ->  ( A  i^i  { 
+oo ,  -oo } )  =  (/) )
13 reldisj 3511 . . . . 5  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( ( A  i^i  {  +oo ,  -oo } )  =  (/)  <->  A  C_  ( ( RR  u.  { 
+oo ,  -oo } ) 
\  {  +oo ,  -oo } ) ) )
14 renfdisj 8901 . . . . . . . 8  |-  ( RR 
i^i  {  +oo ,  -oo } )  =  (/)
15 disj3 3512 . . . . . . . 8  |-  ( ( RR  i^i  {  +oo , 
-oo } )  =  (/)  <->  RR  =  ( RR  \  {  +oo ,  -oo }
) )
1614, 15mpbi 199 . . . . . . 7  |-  RR  =  ( RR  \  {  +oo , 
-oo } )
17 difun2 3546 . . . . . . 7  |-  ( ( RR  u.  {  +oo , 
-oo } )  \  {  +oo ,  -oo } )  =  ( RR  \  {  +oo ,  -oo }
)
1816, 17eqtr4i 2319 . . . . . 6  |-  RR  =  ( ( RR  u.  { 
+oo ,  -oo } ) 
\  {  +oo ,  -oo } )
1918sseq2i 3216 . . . . 5  |-  ( A 
C_  RR  <->  A  C_  ( ( RR  u.  {  +oo , 
-oo } )  \  {  +oo ,  -oo } ) )
2013, 19syl6bbr 254 . . . 4  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( ( A  i^i  {  +oo ,  -oo } )  =  (/)  <->  A  C_  RR ) )
2112, 20syl5ib 210 . . 3  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( -.  (  +oo  e.  A  \/  -oo  e.  A )  ->  A  C_  RR ) )
2221orrd 367 . 2  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( (  +oo  e.  A  \/  -oo  e.  A )  \/  A  C_  RR ) )
23 df-xr 8887 . . 3  |-  RR*  =  ( RR  u.  {  +oo , 
-oo } )
2423sseq2i 3216 . 2  |-  ( A 
C_  RR*  <->  A  C_  ( RR  u.  {  +oo ,  -oo } ) )
25 3orrot 940 . . 3  |-  ( ( A  C_  RR  \/  +oo 
e.  A  \/  -oo  e.  A )  <->  (  +oo  e.  A  \/  -oo  e.  A  \/  A  C_  RR ) )
26 df-3or 935 . . 3  |-  ( ( 
+oo  e.  A  \/  -oo 
e.  A  \/  A  C_  RR )  <->  ( (  +oo  e.  A  \/  -oo  e.  A )  \/  A  C_  RR ) )
2725, 26bitri 240 . 2  |-  ( ( A  C_  RR  \/  +oo 
e.  A  \/  -oo  e.  A )  <->  ( (  +oo  e.  A  \/  -oo  e.  A )  \/  A  C_  RR ) )
2822, 24, 273imtr4i 257 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 357    /\ wa 358    \/ w3o 933    = wceq 1632    e. wcel 1696    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   {cpr 3654   RRcr 8752    +oocpnf 8880    -oocmnf 8881   RR*cxr 8882
This theorem is referenced by:  xrsupss  10643  xrinfmss  10644
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887
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