MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ltrelxr Structured version   Unicode version

Theorem ltrelxr 9680
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 9665 . 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 978 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 4461 . . . . 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 4900 . . . . 5  |-  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) }  C_  ( RR  X.  RR )
53, 4eqsstri 3474 . . . 4  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  C_  ( RR  X.  RR )
6 rexpssxrxp 9670 . . . 4  |-  ( RR 
X.  RR )  C_  ( RR*  X.  RR* )
75, 6sstri 3453 . . 3  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  C_  ( RR*  X.  RR* )
8 ressxr 9669 . . . . . 6  |-  RR  C_  RR*
9 snsspr2 4124 . . . . . . 7  |-  { -oo } 
C_  { +oo , -oo }
10 ssun2 3609 . . . . . . . 8  |-  { +oo , -oo }  C_  ( RR  u.  { +oo , -oo } )
11 df-xr 9664 . . . . . . . 8  |-  RR*  =  ( RR  u.  { +oo , -oo } )
1210, 11sseqtr4i 3477 . . . . . . 7  |-  { +oo , -oo }  C_  RR*
139, 12sstri 3453 . . . . . 6  |-  { -oo } 
C_  RR*
148, 13unssi 3620 . . . . 5  |-  ( RR  u.  { -oo }
)  C_  RR*
15 snsspr1 4123 . . . . . 6  |-  { +oo } 
C_  { +oo , -oo }
1615, 12sstri 3453 . . . . 5  |-  { +oo } 
C_  RR*
17 xpss12 4931 . . . . 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 4931 . . . . 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 3620 . . 3  |-  ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) )  C_  ( RR*  X.  RR* )
227, 21unssi 3620 . 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 3474 1  |-  <  C_  ( RR*  X.  RR* )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    /\ w3a 976    e. wcel 1844    u. cun 3414    C_ wss 3416   {csn 3974   {cpr 3976   class class class wbr 4397   {copab 4454    X. cxp 4823   RRcr 9523    <RR cltrr 9528   +oocpnf 9657   -oocmnf 9658   RR*cxr 9659    < clt 9660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-v 3063  df-un 3421  df-in 3423  df-ss 3430  df-pr 3977  df-opab 4456  df-xp 4831  df-xr 9664  df-ltxr 9665
This theorem is referenced by:  ltrel  9681  dfle2  11408  dflt2  11409  itg2gt0cn  31456
  Copyright terms: Public domain W3C validator