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Theorem ltxr 11349
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 4461 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <RR  y  <->  A  <RR  B ) )
2 df-3an 975 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 4521 . . . . 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 5084 . . . 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 4504 . . . 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 5039 . . . . . . 7  |-  ( A ( ( RR  u.  { -oo } )  X. 
{ +oo } ) B  <-> 
( A  e.  ( RR  u.  { -oo } )  /\  B  e. 
{ +oo } ) )
8 elun 3641 . . . . . . . . . . 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 4055 . . . . . . . . . . 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 4055 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  e.  { +oo }  <->  B  = +oo ) )
1513, 14bi2anan9 873 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  ( RR  u.  { -oo } )  /\  B  e. 
{ +oo } )  <->  ( ( A  = -oo  \/  A  e.  RR )  /\  B  = +oo ) ) )
16 andir 868 . . . . . . . 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 5039 . . . . . . 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 9650 . . . 4  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { -oo } )  X. 
{ +oo } )  u.  ( { -oo }  X.  RR ) ) )
2928breqi 4462 . . 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 4504 . . 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 973    = wceq 1395    e. wcel 1819    u. cun 3469   {csn 4032   class class class wbr 4456   {copab 4514    X. cxp 5006   RRcr 9508    <RR cltrr 9513   +oocpnf 9642   -oocmnf 9643   RR*cxr 9644    < clt 9645
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-ltxr 9650
This theorem is referenced by:  xrltnr  11355  ltpnf  11356  mnflt  11358  mnfltpnf  11360  pnfnlt  11362  nltmnf  11363
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