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Theorem ssxr 8892
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 3647 . . . . . . 7  |-  {  +oo , 
-oo }  =  ( {  +oo }  u.  {  -oo } )
21ineq2i 3367 . . . . . 6  |-  ( A  i^i  {  +oo ,  -oo } )  =  ( A  i^i  ( { 
+oo }  u.  {  -oo } ) )
3 indi 3415 . . . . . 6  |-  ( A  i^i  ( {  +oo }  u.  {  -oo }
) )  =  ( ( A  i^i  {  +oo } )  u.  ( A  i^i  {  -oo }
) )
42, 3eqtri 2303 . . . . 5  |-  ( A  i^i  {  +oo ,  -oo } )  =  ( ( A  i^i  {  +oo } )  u.  ( A  i^i  {  -oo }
) )
5 disjsn 3693 . . . . . . . 8  |-  ( ( A  i^i  {  +oo } )  =  (/)  <->  -.  +oo  e.  A )
6 disjsn 3693 . . . . . . . 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 3490 . . . . . 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 2327 . . . 4  |-  ( -.  (  +oo  e.  A  \/  -oo  e.  A )  ->  ( A  i^i  { 
+oo ,  -oo } )  =  (/) )
13 reldisj 3498 . . . . 5  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( ( A  i^i  {  +oo ,  -oo } )  =  (/)  <->  A  C_  ( ( RR  u.  { 
+oo ,  -oo } ) 
\  {  +oo ,  -oo } ) ) )
14 renfdisj 8885 . . . . . . . 8  |-  ( RR 
i^i  {  +oo ,  -oo } )  =  (/)
15 disj3 3499 . . . . . . . 8  |-  ( ( RR  i^i  {  +oo , 
-oo } )  =  (/)  <->  RR  =  ( RR  \  {  +oo ,  -oo }
) )
1614, 15mpbi 199 . . . . . . 7  |-  RR  =  ( RR  \  {  +oo , 
-oo } )
17 difun2 3533 . . . . . . 7  |-  ( ( RR  u.  {  +oo , 
-oo } )  \  {  +oo ,  -oo } )  =  ( RR  \  {  +oo ,  -oo }
)
1816, 17eqtr4i 2306 . . . . . 6  |-  RR  =  ( ( RR  u.  { 
+oo ,  -oo } ) 
\  {  +oo ,  -oo } )
1918sseq2i 3203 . . . . 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 8871 . . 3  |-  RR*  =  ( RR  u.  {  +oo , 
-oo } )
2423sseq2i 3203 . 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 1623    e. wcel 1684    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   {csn 3640   {cpr 3641   RRcr 8736    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866
This theorem is referenced by:  xrsupss  10627  xrinfmss  10628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871
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