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Theorem ltxr 9585
Description: The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
ltxr  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  ( (
( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  ( A  = -oo  /\  B  = +oo ) )  \/  ( ( A  e.  RR  /\  B  = +oo )  \/  ( A  = -oo  /\  B  e.  RR ) ) ) ) )

Proof of Theorem ltxr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq12 3937 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <RR  y  <->  A  <RR  B ) )
2 df-3an 964 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 3998 . . . . 5  |-  { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  =  { <. x ,  y
>.  |  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) }
41, 3brab2ga 4617 . . . 4  |-  ( A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  <->  ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B ) )
54a1i 9 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  <->  ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B ) ) )
6 brun 3982 . . . 4  |-  ( A ( ( ( RR  u.  { -oo }
)  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) B  <->  ( A
( ( RR  u.  { -oo } )  X. 
{ +oo } ) B  \/  A ( { -oo }  X.  RR ) B ) )
7 brxp 4573 . . . . . . 7  |-  ( A ( ( RR  u.  { -oo } )  X. 
{ +oo } ) B  <-> 
( A  e.  ( RR  u.  { -oo } )  /\  B  e. 
{ +oo } ) )
8 elun 3217 . . . . . . . . . . 11  |-  ( A  e.  ( RR  u.  { -oo } )  <->  ( A  e.  RR  \/  A  e. 
{ -oo } ) )
9 orcom 717 . . . . . . . . . . 11  |-  ( ( A  e.  RR  \/  A  e.  { -oo }
)  <->  ( A  e. 
{ -oo }  \/  A  e.  RR ) )
108, 9bitri 183 . . . . . . . . . 10  |-  ( A  e.  ( RR  u.  { -oo } )  <->  ( A  e.  { -oo }  \/  A  e.  RR )
)
11 elsng 3542 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A  e.  { -oo }  <->  A  = -oo ) )
1211orbi1d 780 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  ( ( A  e.  { -oo }  \/  A  e.  RR ) 
<->  ( A  = -oo  \/  A  e.  RR ) ) )
1310, 12syl5bb 191 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( A  e.  ( RR  u.  { -oo } )  <->  ( A  = -oo  \/  A  e.  RR ) ) )
14 elsng 3542 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  e.  { +oo }  <->  B  = +oo ) )
1513, 14bi2anan9 595 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  ( RR  u.  { -oo } )  /\  B  e. 
{ +oo } )  <->  ( ( A  = -oo  \/  A  e.  RR )  /\  B  = +oo ) ) )
16 andir 808 . . . . . . . 8  |-  ( ( ( A  = -oo  \/  A  e.  RR )  /\  B  = +oo ) 
<->  ( ( A  = -oo  /\  B  = +oo )  \/  ( A  e.  RR  /\  B  = +oo ) ) )
1715, 16syl6bb 195 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  ( RR  u.  { -oo } )  /\  B  e. 
{ +oo } )  <->  ( ( A  = -oo  /\  B  = +oo )  \/  ( A  e.  RR  /\  B  = +oo ) ) ) )
187, 17syl5bb 191 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A ( ( RR  u.  { -oo }
)  X.  { +oo } ) B  <->  ( ( A  = -oo  /\  B  = +oo )  \/  ( A  e.  RR  /\  B  = +oo ) ) ) )
19 brxp 4573 . . . . . . 7  |-  ( A ( { -oo }  X.  RR ) B  <->  ( A  e.  { -oo }  /\  B  e.  RR )
)
2011anbi1d 460 . . . . . . . 8  |-  ( A  e.  RR*  ->  ( ( A  e.  { -oo }  /\  B  e.  RR ) 
<->  ( A  = -oo  /\  B  e.  RR ) ) )
2120adantr 274 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  { -oo }  /\  B  e.  RR )  <->  ( A  = -oo  /\  B  e.  RR ) ) )
2219, 21syl5bb 191 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A ( { -oo }  X.  RR ) B  <-> 
( A  = -oo  /\  B  e.  RR ) ) )
2318, 22orbi12d 782 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A ( ( RR  u.  { -oo } )  X.  { +oo } ) B  \/  A
( { -oo }  X.  RR ) B )  <-> 
( ( ( A  = -oo  /\  B  = +oo )  \/  ( A  e.  RR  /\  B  = +oo ) )  \/  ( A  = -oo  /\  B  e.  RR ) ) ) )
24 orass 756 . . . . 5  |-  ( ( ( ( A  = -oo  /\  B  = +oo )  \/  ( A  e.  RR  /\  B  = +oo ) )  \/  ( A  = -oo  /\  B  e.  RR ) )  <->  ( ( A  = -oo  /\  B  = +oo )  \/  (
( A  e.  RR  /\  B  = +oo )  \/  ( A  = -oo  /\  B  e.  RR ) ) ) )
2523, 24syl6bb 195 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A ( ( RR  u.  { -oo } )  X.  { +oo } ) B  \/  A
( { -oo }  X.  RR ) B )  <-> 
( ( A  = -oo  /\  B  = +oo )  \/  (
( A  e.  RR  /\  B  = +oo )  \/  ( A  = -oo  /\  B  e.  RR ) ) ) ) )
266, 25syl5bb 191 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) B  <->  ( ( A  = -oo  /\  B  = +oo )  \/  (
( A  e.  RR  /\  B  = +oo )  \/  ( A  = -oo  /\  B  e.  RR ) ) ) ) )
275, 26orbi12d 782 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  \/  A ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) B )  <-> 
( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  (
( A  = -oo  /\  B  = +oo )  \/  ( ( A  e.  RR  /\  B  = +oo )  \/  ( A  = -oo  /\  B  e.  RR ) ) ) ) ) )
28 df-ltxr 7824 . . . 4  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { -oo } )  X. 
{ +oo } )  u.  ( { -oo }  X.  RR ) ) )
2928breqi 3938 . . 3  |-  ( A  <  B  <->  A ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { -oo } )  X. 
{ +oo } )  u.  ( { -oo }  X.  RR ) ) ) B )
30 brun 3982 . . 3  |-  ( A ( { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  ( ( ( RR  u.  { -oo }
)  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) ) B  <-> 
( A { <. x ,  y >.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  \/  A ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) B ) )
3129, 30bitri 183 . 2  |-  ( A  <  B  <->  ( A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  \/  A ( ( ( RR  u.  { -oo } )  X.  { +oo } )  u.  ( { -oo }  X.  RR ) ) B ) )
32 orass 756 . 2  |-  ( ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  ( A  = -oo  /\  B  = +oo ) )  \/  ( ( A  e.  RR  /\  B  = +oo )  \/  ( A  = -oo  /\  B  e.  RR ) ) )  <-> 
( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  (
( A  = -oo  /\  B  = +oo )  \/  ( ( A  e.  RR  /\  B  = +oo )  \/  ( A  = -oo  /\  B  e.  RR ) ) ) ) )
3327, 31, 323bitr4g 222 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  B  <->  ( (
( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B )  \/  ( A  = -oo  /\  B  = +oo ) )  \/  ( ( A  e.  RR  /\  B  = +oo )  \/  ( A  = -oo  /\  B  e.  RR ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    = wceq 1331    e. wcel 1480    u. cun 3069   {csn 3527   class class class wbr 3932   {copab 3991    X. cxp 4540   RRcr 7638    <RR cltrr 7643   +oocpnf 7816   -oocmnf 7817   RR*cxr 7818    < clt 7819
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-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3933  df-opab 3993  df-xp 4548  df-ltxr 7824
This theorem is referenced by:  xrltnr  9589  ltpnf  9590  mnflt  9592  mnfltpnf  9594  pnfnlt  9596  nltmnf  9597
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