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Theorem ltrelxr 9660
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 9645 . 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 975 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 4517 . . . . 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 5080 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) }  C_  ( RR  X.  RR )
53, 4eqsstri 3539 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  C_  ( RR  X.  RR )
6 rexpssxrxp 9650 . . . 4  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
75, 6sstri 3518 . . 3  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  C_  ( RR*  X.  RR* )
8 ressxr 9649 . . . . . 6  |-  RR  C_  RR*
9 snsspr2 4183 . . . . . . 7  |-  { -oo } 
C_  { +oo , -oo }
10 ssun2 3673 . . . . . . . 8  |-  { +oo , -oo }  C_  ( RR  u.  { +oo , -oo } )
11 df-xr 9644 . . . . . . . 8  |-  RR*  =  ( RR  u.  { +oo , -oo } )
1210, 11sseqtr4i 3542 . . . . . . 7  |-  { +oo , -oo }  C_  RR*
139, 12sstri 3518 . . . . . 6  |-  { -oo } 
C_  RR*
148, 13unssi 3684 . . . . 5  |-  ( RR  u.  { -oo }
)  C_  RR*
15 snsspr1 4182 . . . . . 6  |-  { +oo } 
C_  { +oo , -oo }
1615, 12sstri 3518 . . . . 5  |-  { +oo } 
C_  RR*
17 xpss12 5114 . . . . 5  |-  ( ( ( RR  u.  { -oo } )  C_  RR*  /\  { +oo }  C_  RR* )  -> 
( ( RR  u.  { -oo } )  X. 
{ +oo } )  C_  ( RR*  X.  RR* )
)
1814, 16, 17mp2an 672 . . . 4  |-  ( ( RR  u.  { -oo } )  X.  { +oo } )  C_  ( RR*  X. 
RR* )
19 xpss12 5114 . . . . 5  |-  ( ( { -oo }  C_  RR* 
/\  RR  C_  RR* )  ->  ( { -oo }  X.  RR )  C_  ( RR*  X.  RR* ) )
2013, 8, 19mp2an 672 . . . 4  |-  ( { -oo }  X.  RR )  C_  ( RR*  X.  RR* )
2118, 20unssi 3684 . . 3  |-  ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) )  C_  ( RR*  X.  RR* )
227, 21unssi 3684 . 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 3539 1  |-  <  C_  ( RR*  X.  RR* )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 973    e. wcel 1767    u. cun 3479    C_ wss 3481   {csn 4033   {cpr 4035   class class class wbr 4453   {copab 4510    X. cxp 5003   RRcr 9503    <RR cltrr 9508   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639    < clt 9640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-un 3486  df-in 3488  df-ss 3495  df-pr 4036  df-opab 4512  df-xp 5011  df-xr 9644  df-ltxr 9645
This theorem is referenced by:  ltrel  9661  dfle2  11365  dflt2  11366  itg2gt0cn  29997
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