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Theorem ssxr 9650
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 4030 . . . . . . 7  |-  { +oo , -oo }  =  ( { +oo }  u.  { -oo } )
21ineq2i 3697 . . . . . 6  |-  ( A  i^i  { +oo , -oo } )  =  ( A  i^i  ( { +oo }  u.  { -oo } ) )
3 indi 3744 . . . . . 6  |-  ( A  i^i  ( { +oo }  u.  { -oo }
) )  =  ( ( A  i^i  { +oo } )  u.  ( A  i^i  { -oo }
) )
42, 3eqtri 2496 . . . . 5  |-  ( A  i^i  { +oo , -oo } )  =  ( ( A  i^i  { +oo } )  u.  ( A  i^i  { -oo }
) )
5 disjsn 4088 . . . . . . . 8  |-  ( ( A  i^i  { +oo } )  =  (/)  <->  -. +oo  e.  A )
6 disjsn 4088 . . . . . . . 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 3862 . . . . . 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 2520 . . . 4  |-  ( -.  ( +oo  e.  A  \/ -oo  e.  A )  ->  ( A  i^i  { +oo , -oo }
)  =  (/) )
13 reldisj 3870 . . . . 5  |-  ( A 
C_  ( RR  u.  { +oo , -oo }
)  ->  ( ( A  i^i  { +oo , -oo } )  =  (/)  <->  A  C_  ( ( RR  u.  { +oo , -oo }
)  \  { +oo , -oo } ) ) )
14 renfdisj 9643 . . . . . . . 8  |-  ( RR 
i^i  { +oo , -oo } )  =  (/)
15 disj3 3871 . . . . . . . 8  |-  ( ( RR  i^i  { +oo , -oo } )  =  (/) 
<->  RR  =  ( RR 
\  { +oo , -oo } ) )
1614, 15mpbi 208 . . . . . . 7  |-  RR  =  ( RR  \  { +oo , -oo } )
17 difun2 3906 . . . . . . 7  |-  ( ( RR  u.  { +oo , -oo } )  \  { +oo , -oo }
)  =  ( RR 
\  { +oo , -oo } )
1816, 17eqtr4i 2499 . . . . . 6  |-  RR  =  ( ( RR  u.  { +oo , -oo }
)  \  { +oo , -oo } )
1918sseq2i 3529 . . . . 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 9628 . . 3  |-  RR*  =  ( RR  u.  { +oo , -oo } )
2423sseq2i 3529 . 2  |-  ( A 
C_  RR*  <->  A  C_  ( RR  u.  { +oo , -oo } ) )
25 3orrot 979 . . 3  |-  ( ( A  C_  RR  \/ +oo  e.  A  \/ -oo  e.  A )  <->  ( +oo  e.  A  \/ -oo  e.  A  \/  A  C_  RR ) )
26 df-3or 974 . . 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 972    = wceq 1379    e. wcel 1767    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   {csn 4027   {cpr 4029   RRcr 9487   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628
This theorem is referenced by:  xrsupss  11496  xrinfmss  11497
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