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Theorem ltxr 11404
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 4422 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <RR  y  <->  A  <RR  B ) )
2 df-3an 984 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 4481 . . . . 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 4921 . . . 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 4465 . . . 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 4876 . . . . . . 7  |-  ( A ( ( RR  u.  { -oo } )  X. 
{ +oo } ) B  <-> 
( A  e.  ( RR  u.  { -oo } )  /\  B  e. 
{ +oo } ) )
8 elun 3603 . . . . . . . . . . 11  |-  ( A  e.  ( RR  u.  { -oo } )  <->  ( A  e.  RR  \/  A  e. 
{ -oo } ) )
9 orcom 388 . . . . . . . . . . 11  |-  ( ( A  e.  RR  \/  A  e.  { -oo }
)  <->  ( A  e. 
{ -oo }  \/  A  e.  RR ) )
108, 9bitri 252 . . . . . . . . . 10  |-  ( A  e.  ( RR  u.  { -oo } )  <->  ( A  e.  { -oo }  \/  A  e.  RR )
)
11 elsncg 4016 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A  e.  { -oo }  <->  A  = -oo ) )
1211orbi1d 707 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  ( ( A  e.  { -oo }  \/  A  e.  RR ) 
<->  ( A  = -oo  \/  A  e.  RR ) ) )
1310, 12syl5bb 260 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( A  e.  ( RR  u.  { -oo } )  <->  ( A  = -oo  \/  A  e.  RR ) ) )
14 elsncg 4016 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  e.  { +oo }  <->  B  = +oo ) )
1513, 14bi2anan9 881 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  ( RR  u.  { -oo } )  /\  B  e. 
{ +oo } )  <->  ( ( A  = -oo  \/  A  e.  RR )  /\  B  = +oo ) ) )
16 andir 876 . . . . . . . 8  |-  ( ( ( A  = -oo  \/  A  e.  RR )  /\  B  = +oo ) 
<->  ( ( A  = -oo  /\  B  = +oo )  \/  ( A  e.  RR  /\  B  = +oo ) ) )
1715, 16syl6bb 264 . . . . . . 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 260 . . . . . 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 4876 . . . . . . 7  |-  ( A ( { -oo }  X.  RR ) B  <->  ( A  e.  { -oo }  /\  B  e.  RR )
)
2011anbi1d 709 . . . . . . . 8  |-  ( A  e.  RR*  ->  ( ( A  e.  { -oo }  /\  B  e.  RR ) 
<->  ( A  = -oo  /\  B  e.  RR ) ) )
2120adantr 466 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  { -oo }  /\  B  e.  RR )  <->  ( A  = -oo  /\  B  e.  RR ) ) )
2219, 21syl5bb 260 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A ( { -oo }  X.  RR ) B  <-> 
( A  = -oo  /\  B  e.  RR ) ) )
2318, 22orbi12d 714 . . . . 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 526 . . . . 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 264 . . . 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 260 . . 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 714 . 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 9669 . . . 4  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { -oo } )  X. 
{ +oo } )  u.  ( { -oo }  X.  RR ) ) )
2928breqi 4423 . . 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 4465 . . 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 252 . 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 526 . 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 291 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 187    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867    u. cun 3431   {csn 3993   class class class wbr 4417   {copab 4474    X. cxp 4843   RRcr 9527    <RR cltrr 9532   +oocpnf 9661   -oocmnf 9662   RR*cxr 9663    < clt 9664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-xp 4851  df-ltxr 9669
This theorem is referenced by:  xrltnr  11410  ltpnf  11411  mnflt  11414  mnfltpnf  11417  pnfnlt  11419  nltmnf  11420
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