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Theorem ssxr 9449
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 3885 . . . . . . 7  |-  { +oo , -oo }  =  ( { +oo }  u.  { -oo } )
21ineq2i 3554 . . . . . 6  |-  ( A  i^i  { +oo , -oo } )  =  ( A  i^i  ( { +oo }  u.  { -oo } ) )
3 indi 3601 . . . . . 6  |-  ( A  i^i  ( { +oo }  u.  { -oo }
) )  =  ( ( A  i^i  { +oo } )  u.  ( A  i^i  { -oo }
) )
42, 3eqtri 2463 . . . . 5  |-  ( A  i^i  { +oo , -oo } )  =  ( ( A  i^i  { +oo } )  u.  ( A  i^i  { -oo }
) )
5 disjsn 3941 . . . . . . . 8  |-  ( ( A  i^i  { +oo } )  =  (/)  <->  -. +oo  e.  A )
6 disjsn 3941 . . . . . . . 8  |-  ( ( A  i^i  { -oo } )  =  (/)  <->  -. -oo  e.  A )
75, 6anbi12i 697 . . . . . . 7  |-  ( ( ( A  i^i  { +oo } )  =  (/)  /\  ( A  i^i  { -oo } )  =  (/) ) 
<->  ( -. +oo  e.  A  /\  -. -oo  e.  A ) )
87biimpri 206 . . . . . 6  |-  ( ( -. +oo  e.  A  /\  -. -oo  e.  A
)  ->  ( ( A  i^i  { +oo }
)  =  (/)  /\  ( A  i^i  { -oo }
)  =  (/) ) )
9 pm4.56 495 . . . . . 6  |-  ( ( -. +oo  e.  A  /\  -. -oo  e.  A
)  <->  -.  ( +oo  e.  A  \/ -oo  e.  A ) )
10 un00 3719 . . . . . 6  |-  ( ( ( A  i^i  { +oo } )  =  (/)  /\  ( A  i^i  { -oo } )  =  (/) ) 
<->  ( ( A  i^i  { +oo } )  u.  ( A  i^i  { -oo } ) )  =  (/) )
118, 9, 103imtr3i 265 . . . . 5  |-  ( -.  ( +oo  e.  A  \/ -oo  e.  A )  ->  ( ( A  i^i  { +oo }
)  u.  ( A  i^i  { -oo }
) )  =  (/) )
124, 11syl5eq 2487 . . . 4  |-  ( -.  ( +oo  e.  A  \/ -oo  e.  A )  ->  ( A  i^i  { +oo , -oo }
)  =  (/) )
13 reldisj 3727 . . . . 5  |-  ( A 
C_  ( RR  u.  { +oo , -oo }
)  ->  ( ( A  i^i  { +oo , -oo } )  =  (/)  <->  A  C_  ( ( RR  u.  { +oo , -oo }
)  \  { +oo , -oo } ) ) )
14 renfdisj 9442 . . . . . . . 8  |-  ( RR 
i^i  { +oo , -oo } )  =  (/)
15 disj3 3728 . . . . . . . 8  |-  ( ( RR  i^i  { +oo , -oo } )  =  (/) 
<->  RR  =  ( RR 
\  { +oo , -oo } ) )
1614, 15mpbi 208 . . . . . . 7  |-  RR  =  ( RR  \  { +oo , -oo } )
17 difun2 3763 . . . . . . 7  |-  ( ( RR  u.  { +oo , -oo } )  \  { +oo , -oo }
)  =  ( RR 
\  { +oo , -oo } )
1816, 17eqtr4i 2466 . . . . . 6  |-  RR  =  ( ( RR  u.  { +oo , -oo }
)  \  { +oo , -oo } )
1918sseq2i 3386 . . . . 5  |-  ( A 
C_  RR  <->  A  C_  ( ( RR  u.  { +oo , -oo } )  \  { +oo , -oo }
) )
2013, 19syl6bbr 263 . . . 4  |-  ( A 
C_  ( RR  u.  { +oo , -oo }
)  ->  ( ( A  i^i  { +oo , -oo } )  =  (/)  <->  A  C_  RR ) )
2112, 20syl5ib 219 . . 3  |-  ( A 
C_  ( RR  u.  { +oo , -oo }
)  ->  ( -.  ( +oo  e.  A  \/ -oo  e.  A )  ->  A  C_  RR ) )
2221orrd 378 . 2  |-  ( A 
C_  ( RR  u.  { +oo , -oo }
)  ->  ( ( +oo  e.  A  \/ -oo  e.  A )  \/  A  C_  RR ) )
23 df-xr 9427 . . 3  |-  RR*  =  ( RR  u.  { +oo , -oo } )
2423sseq2i 3386 . 2  |-  ( A 
C_  RR*  <->  A  C_  ( RR  u.  { +oo , -oo } ) )
25 3orrot 971 . . 3  |-  ( ( A  C_  RR  \/ +oo  e.  A  \/ -oo  e.  A )  <->  ( +oo  e.  A  \/ -oo  e.  A  \/  A  C_  RR ) )
26 df-3or 966 . . 3  |-  ( ( +oo  e.  A  \/ -oo  e.  A  \/  A  C_  RR )  <->  ( ( +oo  e.  A  \/ -oo  e.  A )  \/  A  C_  RR ) )
2725, 26bitri 249 . 2  |-  ( ( A  C_  RR  \/ +oo  e.  A  \/ -oo  e.  A )  <->  ( ( +oo  e.  A  \/ -oo  e.  A )  \/  A  C_  RR ) )
2822, 24, 273imtr4i 266 1  |-  ( A 
C_  RR*  ->  ( A  C_  RR  \/ +oo  e.  A  \/ -oo  e.  A
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    \/ w3o 964    = wceq 1369    e. wcel 1756    \ cdif 3330    u. cun 3331    i^i cin 3332    C_ wss 3333   (/)c0 3642   {csn 3882   {cpr 3884   RRcr 9286   +oocpnf 9420   -oocmnf 9421   RR*cxr 9422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427
This theorem is referenced by:  xrsupss  11276  xrinfmss  11277
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