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Theorem ltxr 10452
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
StepHypRef Expression
1 breq12 4029 . . . . 5  |-  ( ( x  =  A  /\  y  =  B )  ->  ( x  <RR  y  <->  A  <RR  B ) )
2 df-3an 941 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y )  <->  ( (
x  e.  RR  /\  y  e.  RR )  /\  x  <RR  y ) )
32opabbii 4084 . . . . 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 4762 . . . 4  |-  ( A { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) } B  <->  ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <RR  B ) )
54a1i 12 . . 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 4070 . . . 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 4719 . . . . . . 7  |-  ( A ( ( RR  u.  { 
-oo } )  X.  {  +oo } ) B  <->  ( A  e.  ( RR  u.  {  -oo } )  /\  B  e.  {  +oo } ) )
8 elun 3317 . . . . . . . . . . 11  |-  ( A  e.  ( RR  u.  { 
-oo } )  <->  ( A  e.  RR  \/  A  e. 
{  -oo } ) )
9 orcom 378 . . . . . . . . . . 11  |-  ( ( A  e.  RR  \/  A  e.  {  -oo }
)  <->  ( A  e. 
{  -oo }  \/  A  e.  RR ) )
108, 9bitri 242 . . . . . . . . . 10  |-  ( A  e.  ( RR  u.  { 
-oo } )  <->  ( A  e.  {  -oo }  \/  A  e.  RR )
)
11 elsncg 3663 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A  e.  {  -oo }  <->  A  =  -oo ) )
1211orbi1d 686 . . . . . . . . . 10  |-  ( A  e.  RR*  ->  ( ( A  e.  {  -oo }  \/  A  e.  RR ) 
<->  ( A  =  -oo  \/  A  e.  RR ) ) )
1310, 12syl5bb 250 . . . . . . . . 9  |-  ( A  e.  RR*  ->  ( A  e.  ( RR  u.  { 
-oo } )  <->  ( A  =  -oo  \/  A  e.  RR ) ) )
14 elsncg 3663 . . . . . . . . 9  |-  ( B  e.  RR*  ->  ( B  e.  {  +oo }  <->  B  =  +oo ) )
1513, 14bi2anan9 848 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  ( RR  u.  {  -oo } )  /\  B  e. 
{  +oo } )  <->  ( ( A  =  -oo  \/  A  e.  RR )  /\  B  =  +oo ) ) )
16 andir 843 . . . . . . . 8  |-  ( ( ( A  =  -oo  \/  A  e.  RR )  /\  B  =  +oo ) 
<->  ( ( A  = 
-oo  /\  B  =  +oo )  \/  ( A  e.  RR  /\  B  =  +oo ) ) )
1715, 16syl6bb 254 . . . . . . 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 250 . . . . . 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 4719 . . . . . . 7  |-  ( A ( {  -oo }  X.  RR ) B  <->  ( A  e.  {  -oo }  /\  B  e.  RR )
)
2011anbi1d 688 . . . . . . . 8  |-  ( A  e.  RR*  ->  ( ( A  e.  {  -oo }  /\  B  e.  RR ) 
<->  ( A  =  -oo  /\  B  e.  RR ) ) )
2120adantr 453 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  e.  {  -oo }  /\  B  e.  RR )  <->  ( A  =  -oo  /\  B  e.  RR ) ) )
2219, 21syl5bb 250 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A ( {  -oo }  X.  RR ) B  <-> 
( A  =  -oo  /\  B  e.  RR ) ) )
2318, 22orbi12d 693 . . . . 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 512 . . . . 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 254 . . . 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 250 . . 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 693 . 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 8867 . . . 4  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { 
-oo } )  X.  {  +oo } )  u.  ( {  -oo }  X.  RR ) ) )
2928breqi 4030 . . 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 4070 . . 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 242 . 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 512 . 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 281 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 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1628    e. wcel 1688    u. cun 3151   {csn 3641   class class class wbr 4024   {copab 4077    X. cxp 4686   RRcr 8731    <RR cltrr 8736    +oocpnf 8859    -oocmnf 8860   RR*cxr 8861    < clt 8862
This theorem is referenced by:  xrltnr  10457  ltpnf  10458  mnflt  10459  mnfltpnf  10460  pnfnlt  10462  nltmnf  10463
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-ltxr 8867
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