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Theorem ltxr 8465
Description: The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
ltxr  RR*  RR*  <  RR  RR  <RR -oo +oo  RR +oo -oo  RR

Proof of Theorem ltxr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 3760 . . . . 5  <RR  <RR
2 df-3an 886 . . . . . 6  RR  RR  <RR  RR  RR  <RR
32opabbii 3815 . . . . 5  { <. ,  >.  |  RR  RR  <RR  }  { <. ,  >.  |  RR  RR  <RR  }
41, 3brab2ga 4358 . . . 4  { <. , 
>.  |  RR  RR  <RR  }  RR  RR  <RR
54a1i 9 . . 3  RR*  RR*  { <. , 
>.  |  RR  RR  <RR  }  RR  RR  <RR
6 brun 3801 . . . 4  RR  u.  { -oo }  X.  { +oo }  u.  { -oo }  X.  RR  RR  u.  { -oo }  X.  { +oo }  { -oo }  X.  RR
7 brxp 4318 . . . . . . 7  RR  u.  { -oo }  X.  { +oo }  RR  u.  { -oo }  { +oo }
8 elun 3078 . . . . . . . . . . 11  RR  u.  { -oo }  RR  { -oo }
9 orcom 646 . . . . . . . . . . 11  RR  { -oo }  { -oo }  RR
108, 9bitri 173 . . . . . . . . . 10  RR  u.  { -oo }  { -oo }  RR
11 elsncg 3389 . . . . . . . . . . 11  RR*  { -oo } -oo
1211orbi1d 704 . . . . . . . . . 10  RR*  { -oo }  RR -oo  RR
1310, 12syl5bb 181 . . . . . . . . 9  RR*  RR  u.  { -oo } -oo  RR
14 elsncg 3389 . . . . . . . . 9  RR*  { +oo } +oo
1513, 14bi2anan9 538 . . . . . . . 8  RR*  RR*  RR  u.  { -oo }  { +oo } -oo  RR +oo
16 andir 731 . . . . . . . 8 -oo  RR +oo -oo +oo  RR +oo
1715, 16syl6bb 185 . . . . . . 7  RR*  RR*  RR  u.  { -oo }  { +oo } -oo +oo  RR +oo
187, 17syl5bb 181 . . . . . 6  RR*  RR*  RR  u.  { -oo }  X.  { +oo } -oo +oo  RR +oo
19 brxp 4318 . . . . . . 7  { -oo }  X.  RR  { -oo }  RR
2011anbi1d 438 . . . . . . . 8  RR*  { -oo }  RR -oo  RR
2120adantr 261 . . . . . . 7  RR*  RR*  { -oo }  RR -oo  RR
2219, 21syl5bb 181 . . . . . 6  RR*  RR*  { -oo }  X.  RR -oo  RR
2318, 22orbi12d 706 . . . . 5  RR*  RR*  RR  u.  { -oo }  X.  { +oo }  { -oo }  X.  RR -oo +oo  RR +oo -oo  RR
24 orass 683 . . . . 5 -oo +oo  RR +oo -oo  RR -oo +oo  RR +oo -oo  RR
2523, 24syl6bb 185 . . . 4  RR*  RR*  RR  u.  { -oo }  X.  { +oo }  { -oo }  X.  RR -oo +oo  RR +oo -oo  RR
266, 25syl5bb 181 . . 3  RR*  RR*  RR  u.  { -oo }  X.  { +oo }  u.  { -oo }  X.  RR -oo +oo  RR +oo -oo  RR
275, 26orbi12d 706 . 2  RR*  RR*  { <. ,  >.  |  RR  RR  <RR  }  RR  u.  { -oo }  X.  { +oo }  u.  { -oo }  X.  RR  RR  RR  <RR -oo +oo  RR +oo -oo  RR
28 df-ltxr 6862 . . . 4  <  { <. , 
>.  |  RR  RR  <RR  }  u.  RR  u.  { -oo }  X.  { +oo }  u.  { -oo }  X.  RR
2928breqi 3761 . . 3  <  { <. , 
>.  |  RR  RR  <RR  }  u.  RR  u.  { -oo }  X.  { +oo }  u.  { -oo }  X.  RR
30 brun 3801 . . 3  { <. ,  >.  |  RR  RR  <RR  }  u.  RR  u.  { -oo }  X.  { +oo }  u.  { -oo }  X.  RR  { <. , 
>.  |  RR  RR  <RR  }  RR  u.  { -oo }  X.  { +oo }  u.  { -oo }  X.  RR
3129, 30bitri 173 . 2  <  { <. , 
>.  |  RR  RR  <RR  }  RR  u.  { -oo }  X.  { +oo }  u.  { -oo }  X.  RR
32 orass 683 . 2  RR  RR  <RR -oo +oo  RR +oo -oo  RR  RR  RR  <RR -oo +oo  RR +oo -oo  RR
3327, 31, 323bitr4g 212 1  RR*  RR*  <  RR  RR  <RR -oo +oo  RR +oo -oo  RR
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wo 628   w3a 884   wceq 1242   wcel 1390    u. cun 2909   {csn 3367   class class class wbr 3755   {copab 3808    X. cxp 4286   RRcr 6710    <RR cltrr 6715   +oocpnf 6854   -oocmnf 6855   RR*cxr 6856    < clt 6857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-ltxr 6862
This theorem is referenced by:  xrltnr  8471  ltpnf  8472  mnflt  8474  mnfltpnf  8476  pnfnlt  8478  nltmnf  8479
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