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Theorem ltxr 10457
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 4028 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <RR  y  <->  A  <RR  B ) )
2 df-3an 936 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 4083 . . . . 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 4763 . . . 4  |-  ( A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  <->  ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B ) )
54a1i 10 . . 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 4069 . . . 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 4720 . . . . . . 7  |-  ( A ( ( RR  u.  { 
-oo } )  X.  {  +oo } ) B  <->  ( A  e.  ( RR  u.  {  -oo } )  /\  B  e.  {  +oo } ) )
8 elun 3316 . . . . . . . . . . 11  |-  ( A  e.  ( RR  u.  { 
-oo } )  <->  ( A  e.  RR  \/  A  e. 
{  -oo } ) )
9 orcom 376 . . . . . . . . . . 11  |-  ( ( A  e.  RR  \/  A  e.  {  -oo }
)  <->  ( A  e. 
{  -oo }  \/  A  e.  RR ) )
108, 9bitri 240 . . . . . . . . . 10  |-  ( A  e.  ( RR  u.  { 
-oo } )  <->  ( A  e.  {  -oo }  \/  A  e.  RR )
)
11 elsncg 3662 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A  e.  {  -oo }  <->  A  =  -oo ) )
1211orbi1d 683 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  ( ( A  e.  {  -oo }  \/  A  e.  RR ) 
<->  ( A  =  -oo  \/  A  e.  RR ) ) )
1310, 12syl5bb 248 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( A  e.  ( RR  u.  { 
-oo } )  <->  ( A  =  -oo  \/  A  e.  RR ) ) )
14 elsncg 3662 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  e.  {  +oo }  <->  B  =  +oo ) )
1513, 14bi2anan9 843 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  ( RR  u.  {  -oo } )  /\  B  e. 
{  +oo } )  <->  ( ( A  =  -oo  \/  A  e.  RR )  /\  B  =  +oo ) ) )
16 andir 838 . . . . . . . 8  |-  ( ( ( A  =  -oo  \/  A  e.  RR )  /\  B  =  +oo ) 
<->  ( ( A  = 
-oo  /\  B  =  +oo )  \/  ( A  e.  RR  /\  B  =  +oo ) ) )
1715, 16syl6bb 252 . . . . . . 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 248 . . . . . 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 4720 . . . . . . 7  |-  ( A ( {  -oo }  X.  RR ) B  <->  ( A  e.  {  -oo }  /\  B  e.  RR )
)
2011anbi1d 685 . . . . . . . 8  |-  ( A  e.  RR*  ->  ( ( A  e.  {  -oo }  /\  B  e.  RR ) 
<->  ( A  =  -oo  /\  B  e.  RR ) ) )
2120adantr 451 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  {  -oo }  /\  B  e.  RR )  <->  ( A  =  -oo  /\  B  e.  RR ) ) )
2219, 21syl5bb 248 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A ( {  -oo }  X.  RR ) B  <-> 
( A  =  -oo  /\  B  e.  RR ) ) )
2318, 22orbi12d 690 . . . . 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 510 . . . . 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 252 . . . 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 248 . . 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 690 . 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 8872 . . . 4  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { 
-oo } )  X.  {  +oo } )  u.  ( {  -oo }  X.  RR ) ) )
2928breqi 4029 . . 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 4069 . . 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 240 . 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 510 . 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 279 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    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    u. cun 3150   {csn 3640   class class class wbr 4023   {copab 4076    X. cxp 4687   RRcr 8736    <RR cltrr 8741    +oocpnf 8864    -oocmnf 8865   RR*cxr 8866    < clt 8867
This theorem is referenced by:  xrltnr  10462  ltpnf  10463  mnflt  10464  mnfltpnf  10465  pnfnlt  10467  nltmnf  10468
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-ltxr 8872
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