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Theorem ltrelxr 7793
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 7773 . 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 949 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 3965 . . . . 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 4583 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) }  C_  ( RR  X.  RR )
53, 4eqsstri 3099 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  C_  ( RR  X.  RR )
6 rexpssxrxp 7778 . . . 4  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
75, 6sstri 3076 . . 3  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  C_  ( RR*  X.  RR* )
8 ressxr 7777 . . . . . 6  |-  RR  C_  RR*
9 snsspr2 3639 . . . . . . 7  |-  { -oo } 
C_  { +oo , -oo }
10 ssun2 3210 . . . . . . . 8  |-  { +oo , -oo }  C_  ( RR  u.  { +oo , -oo } )
11 df-xr 7772 . . . . . . . 8  |-  RR*  =  ( RR  u.  { +oo , -oo } )
1210, 11sseqtrri 3102 . . . . . . 7  |-  { +oo , -oo }  C_  RR*
139, 12sstri 3076 . . . . . 6  |-  { -oo } 
C_  RR*
148, 13unssi 3221 . . . . 5  |-  ( RR  u.  { -oo }
)  C_  RR*
15 snsspr1 3638 . . . . . 6  |-  { +oo } 
C_  { +oo , -oo }
1615, 12sstri 3076 . . . . 5  |-  { +oo } 
C_  RR*
17 xpss12 4616 . . . . 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 4616 . . . . 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 3221 . . 3  |-  ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) )  C_  ( RR*  X.  RR* )
227, 21unssi 3221 . 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 3099 1  |-  <  C_  ( RR*  X.  RR* )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    /\ w3a 947    e. wcel 1465    u. cun 3039    C_ wss 3041   {csn 3497   {cpr 3498   class class class wbr 3899   {copab 3958    X. cxp 4507   RRcr 7587    <RR cltrr 7592   +oocpnf 7765   -oocmnf 7766   RR*cxr 7767    < clt 7768
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pr 3504  df-opab 3960  df-xp 4515  df-xr 7772  df-ltxr 7773
This theorem is referenced by:  ltrel  7794
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