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Theorem ltxr 11114
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 4316 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <RR  y  <->  A  <RR  B ) )
2 df-3an 967 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 4375 . . . . 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 4931 . . . 4  |-  ( A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  <->  ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B ) )
54a1i 11 . . 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 4359 . . . 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 4889 . . . . . . 7  |-  ( A ( ( RR  u.  { -oo } )  X. 
{ +oo } ) B  <-> 
( A  e.  ( RR  u.  { -oo } )  /\  B  e. 
{ +oo } ) )
8 elun 3516 . . . . . . . . . . 11  |-  ( A  e.  ( RR  u.  { -oo } )  <->  ( A  e.  RR  \/  A  e. 
{ -oo } ) )
9 orcom 387 . . . . . . . . . . 11  |-  ( ( A  e.  RR  \/  A  e.  { -oo }
)  <->  ( A  e. 
{ -oo }  \/  A  e.  RR ) )
108, 9bitri 249 . . . . . . . . . 10  |-  ( A  e.  ( RR  u.  { -oo } )  <->  ( A  e.  { -oo }  \/  A  e.  RR )
)
11 elsncg 3919 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A  e.  { -oo }  <->  A  = -oo ) )
1211orbi1d 702 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  ( ( A  e.  { -oo }  \/  A  e.  RR ) 
<->  ( A  = -oo  \/  A  e.  RR ) ) )
1310, 12syl5bb 257 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( A  e.  ( RR  u.  { -oo } )  <->  ( A  = -oo  \/  A  e.  RR ) ) )
14 elsncg 3919 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  e.  { +oo }  <->  B  = +oo ) )
1513, 14bi2anan9 868 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  ( RR  u.  { -oo } )  /\  B  e. 
{ +oo } )  <->  ( ( A  = -oo  \/  A  e.  RR )  /\  B  = +oo ) ) )
16 andir 863 . . . . . . . 8  |-  ( ( ( A  = -oo  \/  A  e.  RR )  /\  B  = +oo ) 
<->  ( ( A  = -oo  /\  B  = +oo )  \/  ( A  e.  RR  /\  B  = +oo ) ) )
1715, 16syl6bb 261 . . . . . . 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 257 . . . . . 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 4889 . . . . . . 7  |-  ( A ( { -oo }  X.  RR ) B  <->  ( A  e.  { -oo }  /\  B  e.  RR )
)
2011anbi1d 704 . . . . . . . 8  |-  ( A  e.  RR*  ->  ( ( A  e.  { -oo }  /\  B  e.  RR ) 
<->  ( A  = -oo  /\  B  e.  RR ) ) )
2120adantr 465 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  { -oo }  /\  B  e.  RR )  <->  ( A  = -oo  /\  B  e.  RR ) ) )
2219, 21syl5bb 257 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A ( { -oo }  X.  RR ) B  <-> 
( A  = -oo  /\  B  e.  RR ) ) )
2318, 22orbi12d 709 . . . . 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 524 . . . . 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 261 . . . 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 257 . . 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 709 . 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 9442 . . . 4  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { -oo } )  X. 
{ +oo } )  u.  ( { -oo }  X.  RR ) ) )
2928breqi 4317 . . 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 4359 . . 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 249 . 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 524 . 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 288 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    u. cun 3345   {csn 3896   class class class wbr 4311   {copab 4368    X. cxp 4857   RRcr 9300    <RR cltrr 9305   +oocpnf 9434   -oocmnf 9435   RR*cxr 9436    < clt 9437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pr 4550
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-br 4312  df-opab 4370  df-xp 4865  df-ltxr 9442
This theorem is referenced by:  xrltnr  11120  ltpnf  11121  mnflt  11123  mnfltpnf  11125  pnfnlt  11127  nltmnf  11128
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