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Theorem ltrelxr 6877
Description: 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)
Assertion
Ref Expression
ltrelxr  <  C_  RR*  X.  RR*

Proof of Theorem ltrelxr
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltxr 6862 . 2  <  { <. , 
>.  |  RR  RR  <RR  }  u.  RR  u.  { -oo }  X.  { +oo }  u.  { -oo }  X.  RR
2 df-3an 886 . . . . . 6  RR  RR  <RR  RR  RR  <RR
32opabbii 3815 . . . . 5  { <. ,  >.  |  RR  RR  <RR  }  { <. ,  >.  |  RR  RR  <RR  }
4 opabssxp 4357 . . . . 5  { <. ,  >.  |  RR  RR  <RR  }  C_  RR  X.  RR
53, 4eqsstri 2969 . . . 4  { <. ,  >.  |  RR  RR  <RR  }  C_  RR  X.  RR
6 rexpssxrxp 6867 . . . 4  RR 
X.  RR  C_  RR*  X.  RR*
75, 6sstri 2948 . . 3  { <. ,  >.  |  RR  RR  <RR  }  C_  RR*  X.  RR*
8 ressxr 6866 . . . . . 6  RR  C_  RR*
9 snsspr2 3504 . . . . . . 7  { -oo } 
C_  { +oo , -oo }
10 ssun2 3101 . . . . . . . 8  { +oo , -oo }  C_  RR  u.  { +oo , -oo }
11 df-xr 6861 . . . . . . . 8  RR*  RR  u.  { +oo , -oo }
1210, 11sseqtr4i 2972 . . . . . . 7  { +oo , -oo }  C_  RR*
139, 12sstri 2948 . . . . . 6  { -oo } 
C_  RR*
148, 13unssi 3112 . . . . 5  RR  u.  { -oo }  C_  RR*
15 snsspr1 3503 . . . . . 6  { +oo } 
C_  { +oo , -oo }
1615, 12sstri 2948 . . . . 5  { +oo } 
C_  RR*
17 xpss12 4388 . . . . 5  RR  u.  { -oo }  C_  RR*  { +oo }  C_  RR*  RR  u.  { -oo }  X.  { +oo }  C_  RR*  X.  RR*
1814, 16, 17mp2an 402 . . . 4  RR  u.  { -oo }  X.  { +oo }  C_  RR*  X. 
RR*
19 xpss12 4388 . . . . 5  { -oo }  C_  RR*  RR  C_  RR*  { -oo }  X.  RR  C_  RR*  X. 
RR*
2013, 8, 19mp2an 402 . . . 4  { -oo }  X.  RR 
C_  RR*  X.  RR*
2118, 20unssi 3112 . . 3  RR  u.  { -oo }  X.  { +oo }  u.  { -oo }  X.  RR  C_  RR*  X. 
RR*
227, 21unssi 3112 . 2  { <. ,  >.  |  RR  RR  <RR  }  u.  RR  u.  { -oo }  X.  { +oo }  u.  { -oo }  X.  RR  C_  RR*  X. 
RR*
231, 22eqsstri 2969 1  <  C_  RR*  X.  RR*
Colors of variables: wff set class
Syntax hints:   wa 97   w3a 884   wcel 1390    u. cun 2909    C_ wss 2911   {csn 3367   {cpr 3368   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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pr 3374  df-opab 3810  df-xp 4294  df-xr 6861  df-ltxr 6862
This theorem is referenced by:  ltrel  6878
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