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Theorem ltxr 10336
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 3925 . . . . 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 3980 . . . . 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 4670 . . . 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 3966 . . . 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 4627 . . . . . . 7  |-  ( A ( ( RR  u.  { 
-oo } )  X.  {  +oo } ) B  <->  ( A  e.  ( RR  u.  {  -oo } )  /\  B  e.  {  +oo } ) )
8 elun 3226 . . . . . . . . . . 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 3566 . . . . . . . . . . 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 3566 . . . . . . . . 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 4627 . . . . . . 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 8752 . . . 4  |-  <  =  ( { <. x ,  y
>.  |  ( x  e.  RR  /\  y  e.  RR  /\  x  <RR  y ) }  u.  (
( ( RR  u.  { 
-oo } )  X.  {  +oo } )  u.  ( {  -oo }  X.  RR ) ) )
2928breqi 3926 . . 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 3966 . . 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 1619    e. wcel 1621    u. cun 3076   {csn 3544   class class class wbr 3920   {copab 3973    X. cxp 4578   RRcr 8616    <RR cltrr 8621    +oocpnf 8744    -oocmnf 8745   RR*cxr 8746    < clt 8747
This theorem is referenced by:  xrltnr  10341  ltpnf  10342  mnflt  10343  mnfltpnf  10344  pnfnlt  10346  nltmnf  10347
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-xp 4594  df-ltxr 8752
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