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Theorem ltrelxr 7825
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 7805 . 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 964 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 3995 . . . . 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 4613 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) }  C_  ( RR  X.  RR )
53, 4eqsstri 3129 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  C_  ( RR  X.  RR )
6 rexpssxrxp 7810 . . . 4  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
75, 6sstri 3106 . . 3  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  C_  ( RR*  X.  RR* )
8 ressxr 7809 . . . . . 6  |-  RR  C_  RR*
9 snsspr2 3669 . . . . . . 7  |-  { -oo } 
C_  { +oo , -oo }
10 ssun2 3240 . . . . . . . 8  |-  { +oo , -oo }  C_  ( RR  u.  { +oo , -oo } )
11 df-xr 7804 . . . . . . . 8  |-  RR*  =  ( RR  u.  { +oo , -oo } )
1210, 11sseqtrri 3132 . . . . . . 7  |-  { +oo , -oo }  C_  RR*
139, 12sstri 3106 . . . . . 6  |-  { -oo } 
C_  RR*
148, 13unssi 3251 . . . . 5  |-  ( RR  u.  { -oo }
)  C_  RR*
15 snsspr1 3668 . . . . . 6  |-  { +oo } 
C_  { +oo , -oo }
1615, 12sstri 3106 . . . . 5  |-  { +oo } 
C_  RR*
17 xpss12 4646 . . . . 5  |-  ( ( ( RR  u.  { -oo } )  C_  RR*  /\  { +oo }  C_  RR* )  -> 
( ( RR  u.  { -oo } )  X. 
{ +oo } )  C_  ( RR*  X.  RR* )
)
1814, 16, 17mp2an 422 . . . 4  |-  ( ( RR  u.  { -oo } )  X.  { +oo } )  C_  ( RR*  X. 
RR* )
19 xpss12 4646 . . . . 5  |-  ( ( { -oo }  C_  RR* 
/\  RR  C_  RR* )  ->  ( { -oo }  X.  RR )  C_  ( RR*  X.  RR* ) )
2013, 8, 19mp2an 422 . . . 4  |-  ( { -oo }  X.  RR )  C_  ( RR*  X.  RR* )
2118, 20unssi 3251 . . 3  |-  ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) )  C_  ( RR*  X.  RR* )
227, 21unssi 3251 . 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 3129 1  |-  <  C_  ( RR*  X.  RR* )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    /\ w3a 962    e. wcel 1480    u. cun 3069    C_ wss 3071   {csn 3527   {cpr 3528   class class class wbr 3929   {copab 3988    X. cxp 4537   RRcr 7619    <RR cltrr 7624   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799    < clt 7800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pr 3534  df-opab 3990  df-xp 4545  df-xr 7804  df-ltxr 7805
This theorem is referenced by:  ltrel  7826
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