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Theorem ssxr 9702
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 4005 . . . . . . 7  |-  { +oo , -oo }  =  ( { +oo }  u.  { -oo } )
21ineq2i 3667 . . . . . 6  |-  ( A  i^i  { +oo , -oo } )  =  ( A  i^i  ( { +oo }  u.  { -oo } ) )
3 indi 3725 . . . . . 6  |-  ( A  i^i  ( { +oo }  u.  { -oo }
) )  =  ( ( A  i^i  { +oo } )  u.  ( A  i^i  { -oo }
) )
42, 3eqtri 2458 . . . . 5  |-  ( A  i^i  { +oo , -oo } )  =  ( ( A  i^i  { +oo } )  u.  ( A  i^i  { -oo }
) )
5 disjsn 4063 . . . . . . . 8  |-  ( ( A  i^i  { +oo } )  =  (/)  <->  -. +oo  e.  A )
6 disjsn 4063 . . . . . . . 8  |-  ( ( A  i^i  { -oo } )  =  (/)  <->  -. -oo  e.  A )
75, 6anbi12i 701 . . . . . . 7  |-  ( ( ( A  i^i  { +oo } )  =  (/)  /\  ( A  i^i  { -oo } )  =  (/) ) 
<->  ( -. +oo  e.  A  /\  -. -oo  e.  A ) )
87biimpri 209 . . . . . 6  |-  ( ( -. +oo  e.  A  /\  -. -oo  e.  A
)  ->  ( ( A  i^i  { +oo }
)  =  (/)  /\  ( A  i^i  { -oo }
)  =  (/) ) )
9 pm4.56 497 . . . . . 6  |-  ( ( -. +oo  e.  A  /\  -. -oo  e.  A
)  <->  -.  ( +oo  e.  A  \/ -oo  e.  A ) )
10 un00 3834 . . . . . 6  |-  ( ( ( A  i^i  { +oo } )  =  (/)  /\  ( A  i^i  { -oo } )  =  (/) ) 
<->  ( ( A  i^i  { +oo } )  u.  ( A  i^i  { -oo } ) )  =  (/) )
118, 9, 103imtr3i 268 . . . . 5  |-  ( -.  ( +oo  e.  A  \/ -oo  e.  A )  ->  ( ( A  i^i  { +oo }
)  u.  ( A  i^i  { -oo }
) )  =  (/) )
124, 11syl5eq 2482 . . . 4  |-  ( -.  ( +oo  e.  A  \/ -oo  e.  A )  ->  ( A  i^i  { +oo , -oo }
)  =  (/) )
13 reldisj 3842 . . . . 5  |-  ( A 
C_  ( RR  u.  { +oo , -oo }
)  ->  ( ( A  i^i  { +oo , -oo } )  =  (/)  <->  A  C_  ( ( RR  u.  { +oo , -oo }
)  \  { +oo , -oo } ) ) )
14 renfdisj 9693 . . . . . . . 8  |-  ( RR 
i^i  { +oo , -oo } )  =  (/)
15 disj3 3843 . . . . . . . 8  |-  ( ( RR  i^i  { +oo , -oo } )  =  (/) 
<->  RR  =  ( RR 
\  { +oo , -oo } ) )
1614, 15mpbi 211 . . . . . . 7  |-  RR  =  ( RR  \  { +oo , -oo } )
17 difun2 3881 . . . . . . 7  |-  ( ( RR  u.  { +oo , -oo } )  \  { +oo , -oo }
)  =  ( RR 
\  { +oo , -oo } )
1816, 17eqtr4i 2461 . . . . . 6  |-  RR  =  ( ( RR  u.  { +oo , -oo }
)  \  { +oo , -oo } )
1918sseq2i 3495 . . . . 5  |-  ( A 
C_  RR  <->  A  C_  ( ( RR  u.  { +oo , -oo } )  \  { +oo , -oo }
) )
2013, 19syl6bbr 266 . . . 4  |-  ( A 
C_  ( RR  u.  { +oo , -oo }
)  ->  ( ( A  i^i  { +oo , -oo } )  =  (/)  <->  A  C_  RR ) )
2112, 20syl5ib 222 . . 3  |-  ( A 
C_  ( RR  u.  { +oo , -oo }
)  ->  ( -.  ( +oo  e.  A  \/ -oo  e.  A )  ->  A  C_  RR ) )
2221orrd 379 . 2  |-  ( A 
C_  ( RR  u.  { +oo , -oo }
)  ->  ( ( +oo  e.  A  \/ -oo  e.  A )  \/  A  C_  RR ) )
23 df-xr 9678 . . 3  |-  RR*  =  ( RR  u.  { +oo , -oo } )
2423sseq2i 3495 . 2  |-  ( A 
C_  RR*  <->  A  C_  ( RR  u.  { +oo , -oo } ) )
25 3orrot 988 . . 3  |-  ( ( A  C_  RR  \/ +oo  e.  A  \/ -oo  e.  A )  <->  ( +oo  e.  A  \/ -oo  e.  A  \/  A  C_  RR ) )
26 df-3or 983 . . 3  |-  ( ( +oo  e.  A  \/ -oo  e.  A  \/  A  C_  RR )  <->  ( ( +oo  e.  A  \/ -oo  e.  A )  \/  A  C_  RR ) )
2725, 26bitri 252 . 2  |-  ( ( A  C_  RR  \/ +oo  e.  A  \/ -oo  e.  A )  <->  ( ( +oo  e.  A  \/ -oo  e.  A )  \/  A  C_  RR ) )
2822, 24, 273imtr4i 269 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 369    /\ wa 370    \/ w3o 981    = wceq 1437    e. wcel 1870    \ cdif 3439    u. cun 3440    i^i cin 3441    C_ wss 3442   (/)c0 3767   {csn 4002   {cpr 4004   RRcr 9537   +oocpnf 9671   -oocmnf 9672   RR*cxr 9673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678
This theorem is referenced by:  xrsupss  11594  xrinfmss  11595
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