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Theorem ltrelxr 7310
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  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ltxr 7290 . 2  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { -oo } )  X. 
{ +oo } )  u.  ( { -oo }  X.  RR ) ) )
2 df-3an 922 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 3865 . . . . 5  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  =  { <. x ,  y
>.  |  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) }
4 opabssxp 4460 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) }  C_  ( RR  X.  RR )
53, 4eqsstri 3038 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  C_  ( RR  X.  RR )
6 rexpssxrxp 7295 . . . 4  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
75, 6sstri 3017 . . 3  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  C_  ( RR*  X.  RR* )
8 ressxr 7294 . . . . . 6  |-  RR  C_  RR*
9 snsspr2 3554 . . . . . . 7  |-  { -oo } 
C_  { +oo , -oo }
10 ssun2 3146 . . . . . . . 8  |-  { +oo , -oo }  C_  ( RR  u.  { +oo , -oo } )
11 df-xr 7289 . . . . . . . 8  |-  RR*  =  ( RR  u.  { +oo , -oo } )
1210, 11sseqtr4i 3041 . . . . . . 7  |-  { +oo , -oo }  C_  RR*
139, 12sstri 3017 . . . . . 6  |-  { -oo } 
C_  RR*
148, 13unssi 3157 . . . . 5  |-  ( RR  u.  { -oo }
)  C_  RR*
15 snsspr1 3553 . . . . . 6  |-  { +oo } 
C_  { +oo , -oo }
1615, 12sstri 3017 . . . . 5  |-  { +oo } 
C_  RR*
17 xpss12 4493 . . . . 5  |-  ( ( ( RR  u.  { -oo } )  C_  RR*  /\  { +oo }  C_  RR* )  -> 
( ( RR  u.  { -oo } )  X. 
{ +oo } )  C_  ( RR*  X.  RR* )
)
1814, 16, 17mp2an 417 . . . 4  |-  ( ( RR  u.  { -oo } )  X.  { +oo } )  C_  ( RR*  X. 
RR* )
19 xpss12 4493 . . . . 5  |-  ( ( { -oo }  C_  RR* 
/\  RR  C_  RR* )  ->  ( { -oo }  X.  RR )  C_  ( RR*  X.  RR* ) )
2013, 8, 19mp2an 417 . . . 4  |-  ( { -oo }  X.  RR )  C_  ( RR*  X.  RR* )
2118, 20unssi 3157 . . 3  |-  ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) )  C_  ( RR*  X.  RR* )
227, 21unssi 3157 . 2  |-  ( {
<. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { -oo } )  X. 
{ +oo } )  u.  ( { -oo }  X.  RR ) ) ) 
C_  ( RR*  X.  RR* )
231, 22eqsstri 3038 1  |-  <  C_  ( RR*  X.  RR* )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    /\ w3a 920    e. wcel 1434    u. cun 2980    C_ wss 2982   {csn 3416   {cpr 3417   class class class wbr 3805   {copab 3858    X. cxp 4389   RRcr 7112    <RR cltrr 7117   +oocpnf 7282   -oocmnf 7283   RR*cxr 7284    < clt 7285
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pr 3423  df-opab 3860  df-xp 4397  df-xr 7289  df-ltxr 7290
This theorem is referenced by:  ltrel  7311
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