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Theorem ltrelxr 9139
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 9125 . 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 938 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 4272 . . . . 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 4950 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) }  C_  ( RR  X.  RR )
53, 4eqsstri 3378 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  C_  ( RR  X.  RR )
6 ressxr 9129 . . . . 5  |-  RR  C_  RR*
7 xpss12 4981 . . . . 5  |-  ( ( RR  C_  RR*  /\  RR  C_ 
RR* )  ->  ( RR  X.  RR )  C_  ( RR*  X.  RR* )
)
86, 6, 7mp2an 654 . . . 4  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
95, 8sstri 3357 . . 3  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  C_  ( RR*  X.  RR* )
10 snsspr2 3948 . . . . . . 7  |-  {  -oo } 
C_  {  +oo ,  -oo }
11 ssun2 3511 . . . . . . . 8  |-  {  +oo , 
-oo }  C_  ( RR  u.  {  +oo ,  -oo } )
12 df-xr 9124 . . . . . . . 8  |-  RR*  =  ( RR  u.  {  +oo , 
-oo } )
1311, 12sseqtr4i 3381 . . . . . . 7  |-  {  +oo , 
-oo }  C_  RR*
1410, 13sstri 3357 . . . . . 6  |-  {  -oo } 
C_  RR*
156, 14unssi 3522 . . . . 5  |-  ( RR  u.  {  -oo }
)  C_  RR*
16 snsspr1 3947 . . . . . 6  |-  {  +oo } 
C_  {  +oo ,  -oo }
1716, 13sstri 3357 . . . . 5  |-  {  +oo } 
C_  RR*
18 xpss12 4981 . . . . 5  |-  ( ( ( RR  u.  {  -oo } )  C_  RR*  /\  {  +oo }  C_  RR* )  -> 
( ( RR  u.  { 
-oo } )  X.  {  +oo } )  C_  ( RR*  X.  RR* ) )
1915, 17, 18mp2an 654 . . . 4  |-  ( ( RR  u.  {  -oo } )  X.  {  +oo } )  C_  ( RR*  X. 
RR* )
20 xpss12 4981 . . . . 5  |-  ( ( {  -oo }  C_  RR* 
/\  RR  C_  RR* )  ->  ( {  -oo }  X.  RR )  C_  ( RR*  X.  RR* ) )
2114, 6, 20mp2an 654 . . . 4  |-  ( { 
-oo }  X.  RR )  C_  ( RR*  X.  RR* )
2219, 21unssi 3522 . . 3  |-  ( ( ( RR  u.  {  -oo } )  X.  {  +oo } )  u.  ( {  -oo }  X.  RR ) )  C_  ( RR*  X.  RR* )
239, 22unssi 3522 . 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* )
241, 23eqsstri 3378 1  |-  <  C_  ( RR*  X.  RR* )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    /\ w3a 936    e. wcel 1725    u. cun 3318    C_ wss 3320   {csn 3814   {cpr 3815   class class class wbr 4212   {copab 4265    X. cxp 4876   RRcr 8989    <RR cltrr 8994    +oocpnf 9117    -oocmnf 9118   RR*cxr 9119    < clt 9120
This theorem is referenced by:  ltrel  9140  dfle2  10740  dflt2  10741  itg2gt0cn  26260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-in 3327  df-ss 3334  df-pr 3821  df-opab 4267  df-xp 4884  df-xr 9124  df-ltxr 9125
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