MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssxr Structured version   Unicode version

Theorem ssxr 9138
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 3814 . . . . . . 7  |-  {  +oo , 
-oo }  =  ( {  +oo }  u.  {  -oo } )
21ineq2i 3532 . . . . . 6  |-  ( A  i^i  {  +oo ,  -oo } )  =  ( A  i^i  ( { 
+oo }  u.  {  -oo } ) )
3 indi 3580 . . . . . 6  |-  ( A  i^i  ( {  +oo }  u.  {  -oo }
) )  =  ( ( A  i^i  {  +oo } )  u.  ( A  i^i  {  -oo }
) )
42, 3eqtri 2456 . . . . 5  |-  ( A  i^i  {  +oo ,  -oo } )  =  ( ( A  i^i  {  +oo } )  u.  ( A  i^i  {  -oo }
) )
5 disjsn 3861 . . . . . . . 8  |-  ( ( A  i^i  {  +oo } )  =  (/)  <->  -.  +oo  e.  A )
6 disjsn 3861 . . . . . . . 8  |-  ( ( A  i^i  {  -oo } )  =  (/)  <->  -.  -oo  e.  A )
75, 6anbi12i 679 . . . . . . 7  |-  ( ( ( A  i^i  {  +oo } )  =  (/)  /\  ( A  i^i  {  -oo } )  =  (/) ) 
<->  ( -.  +oo  e.  A  /\  -.  -oo  e.  A ) )
87biimpri 198 . . . . . 6  |-  ( ( -.  +oo  e.  A  /\  -.  -oo  e.  A
)  ->  ( ( A  i^i  {  +oo }
)  =  (/)  /\  ( A  i^i  {  -oo }
)  =  (/) ) )
9 pm4.56 482 . . . . . 6  |-  ( ( -.  +oo  e.  A  /\  -.  -oo  e.  A
)  <->  -.  (  +oo  e.  A  \/  -oo  e.  A ) )
10 un00 3656 . . . . . 6  |-  ( ( ( A  i^i  {  +oo } )  =  (/)  /\  ( A  i^i  {  -oo } )  =  (/) ) 
<->  ( ( A  i^i  { 
+oo } )  u.  ( A  i^i  {  -oo }
) )  =  (/) )
118, 9, 103imtr3i 257 . . . . 5  |-  ( -.  (  +oo  e.  A  \/  -oo  e.  A )  ->  ( ( A  i^i  {  +oo }
)  u.  ( A  i^i  {  -oo }
) )  =  (/) )
124, 11syl5eq 2480 . . . 4  |-  ( -.  (  +oo  e.  A  \/  -oo  e.  A )  ->  ( A  i^i  { 
+oo ,  -oo } )  =  (/) )
13 reldisj 3664 . . . . 5  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( ( A  i^i  {  +oo ,  -oo } )  =  (/)  <->  A  C_  ( ( RR  u.  { 
+oo ,  -oo } ) 
\  {  +oo ,  -oo } ) ) )
14 renfdisj 9131 . . . . . . . 8  |-  ( RR 
i^i  {  +oo ,  -oo } )  =  (/)
15 disj3 3665 . . . . . . . 8  |-  ( ( RR  i^i  {  +oo , 
-oo } )  =  (/)  <->  RR  =  ( RR  \  {  +oo ,  -oo }
) )
1614, 15mpbi 200 . . . . . . 7  |-  RR  =  ( RR  \  {  +oo , 
-oo } )
17 difun2 3700 . . . . . . 7  |-  ( ( RR  u.  {  +oo , 
-oo } )  \  {  +oo ,  -oo } )  =  ( RR  \  {  +oo ,  -oo }
)
1816, 17eqtr4i 2459 . . . . . 6  |-  RR  =  ( ( RR  u.  { 
+oo ,  -oo } ) 
\  {  +oo ,  -oo } )
1918sseq2i 3366 . . . . 5  |-  ( A 
C_  RR  <->  A  C_  ( ( RR  u.  {  +oo , 
-oo } )  \  {  +oo ,  -oo } ) )
2013, 19syl6bbr 255 . . . 4  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( ( A  i^i  {  +oo ,  -oo } )  =  (/)  <->  A  C_  RR ) )
2112, 20syl5ib 211 . . 3  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( -.  (  +oo  e.  A  \/  -oo  e.  A )  ->  A  C_  RR ) )
2221orrd 368 . 2  |-  ( A 
C_  ( RR  u.  { 
+oo ,  -oo } )  ->  ( (  +oo  e.  A  \/  -oo  e.  A )  \/  A  C_  RR ) )
23 df-xr 9117 . . 3  |-  RR*  =  ( RR  u.  {  +oo , 
-oo } )
2423sseq2i 3366 . 2  |-  ( A 
C_  RR*  <->  A  C_  ( RR  u.  {  +oo ,  -oo } ) )
25 3orrot 942 . . 3  |-  ( ( A  C_  RR  \/  +oo 
e.  A  \/  -oo  e.  A )  <->  (  +oo  e.  A  \/  -oo  e.  A  \/  A  C_  RR ) )
26 df-3or 937 . . 3  |-  ( ( 
+oo  e.  A  \/  -oo 
e.  A  \/  A  C_  RR )  <->  ( (  +oo  e.  A  \/  -oo  e.  A )  \/  A  C_  RR ) )
2725, 26bitri 241 . 2  |-  ( ( A  C_  RR  \/  +oo 
e.  A  \/  -oo  e.  A )  <->  ( (  +oo  e.  A  \/  -oo  e.  A )  \/  A  C_  RR ) )
2822, 24, 273imtr4i 258 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 358    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725    \ cdif 3310    u. cun 3311    i^i cin 3312    C_ wss 3313   (/)c0 3621   {csn 3807   {cpr 3808   RRcr 8982    +oocpnf 9110    -oocmnf 9111   RR*cxr 9112
This theorem is referenced by:  xrsupss  10880  xrinfmss  10881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694  ax-resscn 9040
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-er 6898  df-en 7103  df-dom 7104  df-sdom 7105  df-pnf 9115  df-mnf 9116  df-xr 9117
  Copyright terms: Public domain W3C validator