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Theorem ordthaus 17406
Description: The order topology of a total order is Hausdorff. (Contributed by Mario Carneiro, 13-Sep-2015.)
Assertion
Ref Expression
ordthaus  |-  ( R  e.  TosetRel  ->  (ordTop `  R )  e.  Haus )

Proof of Theorem ordthaus
Dummy variables  m  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2408 . . . . . 6  |-  dom  R  =  dom  R
21ordthauslem 17405 . . . . 5  |-  ( ( R  e.  TosetRel  /\  x  e.  dom  R  /\  y  e.  dom  R )  -> 
( x R y  ->  ( x  =/=  y  ->  E. m  e.  (ordTop `  R ) E. n  e.  (ordTop `  R ) ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) ) ) )
31ordthauslem 17405 . . . . . . 7  |-  ( ( R  e.  TosetRel  /\  y  e.  dom  R  /\  x  e.  dom  R )  -> 
( y R x  ->  ( y  =/=  x  ->  E. n  e.  (ordTop `  R ) E. m  e.  (ordTop `  R ) ( y  e.  n  /\  x  e.  m  /\  (
n  i^i  m )  =  (/) ) ) ) )
4 necom 2652 . . . . . . . 8  |-  ( y  =/=  x  <->  x  =/=  y )
5 3ancoma 943 . . . . . . . . . . 11  |-  ( ( y  e.  n  /\  x  e.  m  /\  ( n  i^i  m
)  =  (/) )  <->  ( x  e.  m  /\  y  e.  n  /\  (
n  i^i  m )  =  (/) ) )
6 incom 3497 . . . . . . . . . . . . 13  |-  ( n  i^i  m )  =  ( m  i^i  n
)
76eqeq1i 2415 . . . . . . . . . . . 12  |-  ( ( n  i^i  m )  =  (/)  <->  ( m  i^i  n )  =  (/) )
873anbi3i 1146 . . . . . . . . . . 11  |-  ( ( x  e.  m  /\  y  e.  n  /\  ( n  i^i  m
)  =  (/) )  <->  ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) )
95, 8bitri 241 . . . . . . . . . 10  |-  ( ( y  e.  n  /\  x  e.  m  /\  ( n  i^i  m
)  =  (/) )  <->  ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) )
1092rexbii 2697 . . . . . . . . 9  |-  ( E. n  e.  (ordTop `  R ) E. m  e.  (ordTop `  R )
( y  e.  n  /\  x  e.  m  /\  ( n  i^i  m
)  =  (/) )  <->  E. n  e.  (ordTop `  R ) E. m  e.  (ordTop `  R ) ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) )
11 rexcom 2833 . . . . . . . . 9  |-  ( E. n  e.  (ordTop `  R ) E. m  e.  (ordTop `  R )
( x  e.  m  /\  y  e.  n  /\  ( m  i^i  n
)  =  (/) )  <->  E. m  e.  (ordTop `  R ) E. n  e.  (ordTop `  R ) ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) )
1210, 11bitri 241 . . . . . . . 8  |-  ( E. n  e.  (ordTop `  R ) E. m  e.  (ordTop `  R )
( y  e.  n  /\  x  e.  m  /\  ( n  i^i  m
)  =  (/) )  <->  E. m  e.  (ordTop `  R ) E. n  e.  (ordTop `  R ) ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) )
134, 12imbi12i 317 . . . . . . 7  |-  ( ( y  =/=  x  ->  E. n  e.  (ordTop `  R ) E. m  e.  (ordTop `  R )
( y  e.  n  /\  x  e.  m  /\  ( n  i^i  m
)  =  (/) ) )  <-> 
( x  =/=  y  ->  E. m  e.  (ordTop `  R ) E. n  e.  (ordTop `  R )
( x  e.  m  /\  y  e.  n  /\  ( m  i^i  n
)  =  (/) ) ) )
143, 13syl6ib 218 . . . . . 6  |-  ( ( R  e.  TosetRel  /\  y  e.  dom  R  /\  x  e.  dom  R )  -> 
( y R x  ->  ( x  =/=  y  ->  E. m  e.  (ordTop `  R ) E. n  e.  (ordTop `  R ) ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) ) ) )
15143com23 1159 . . . . 5  |-  ( ( R  e.  TosetRel  /\  x  e.  dom  R  /\  y  e.  dom  R )  -> 
( y R x  ->  ( x  =/=  y  ->  E. m  e.  (ordTop `  R ) E. n  e.  (ordTop `  R ) ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) ) ) )
161tsrlin 14610 . . . . 5  |-  ( ( R  e.  TosetRel  /\  x  e.  dom  R  /\  y  e.  dom  R )  -> 
( x R y  \/  y R x ) )
172, 15, 16mpjaod 371 . . . 4  |-  ( ( R  e.  TosetRel  /\  x  e.  dom  R  /\  y  e.  dom  R )  -> 
( x  =/=  y  ->  E. m  e.  (ordTop `  R ) E. n  e.  (ordTop `  R )
( x  e.  m  /\  y  e.  n  /\  ( m  i^i  n
)  =  (/) ) ) )
18173expb 1154 . . 3  |-  ( ( R  e.  TosetRel  /\  (
x  e.  dom  R  /\  y  e.  dom  R ) )  ->  (
x  =/=  y  ->  E. m  e.  (ordTop `  R ) E. n  e.  (ordTop `  R )
( x  e.  m  /\  y  e.  n  /\  ( m  i^i  n
)  =  (/) ) ) )
1918ralrimivva 2762 . 2  |-  ( R  e.  TosetRel  ->  A. x  e.  dom  R A. y  e.  dom  R ( x  =/=  y  ->  E. m  e.  (ordTop `  R ) E. n  e.  (ordTop `  R )
( x  e.  m  /\  y  e.  n  /\  ( m  i^i  n
)  =  (/) ) ) )
201ordttopon 17215 . . 3  |-  ( R  e.  TosetRel  ->  (ordTop `  R )  e.  (TopOn `  dom  R ) )
21 ishaus2 17373 . . 3  |-  ( (ordTop `  R )  e.  (TopOn `  dom  R )  -> 
( (ordTop `  R
)  e.  Haus  <->  A. x  e.  dom  R A. y  e.  dom  R ( x  =/=  y  ->  E. m  e.  (ordTop `  R ) E. n  e.  (ordTop `  R ) ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) ) ) )
2220, 21syl 16 . 2  |-  ( R  e.  TosetRel  ->  ( (ordTop `  R )  e.  Haus  <->  A. x  e.  dom  R A. y  e.  dom  R ( x  =/=  y  ->  E. m  e.  (ordTop `  R ) E. n  e.  (ordTop `  R )
( x  e.  m  /\  y  e.  n  /\  ( m  i^i  n
)  =  (/) ) ) ) )
2319, 22mpbird 224 1  |-  ( R  e.  TosetRel  ->  (ordTop `  R )  e.  Haus )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2571   A.wral 2670   E.wrex 2671    i^i cin 3283   (/)c0 3592   class class class wbr 4176   dom cdm 4841   ` cfv 5417  ordTopcordt 13680    TosetRel ctsr 14584  TopOnctopon 16918   Hauscha 17330
This theorem is referenced by:  xrge0tsms  18822  xrhaus  24085  xrge0tsmsd  24180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-recs 6596  df-rdg 6631  df-1o 6687  df-oadd 6691  df-er 6868  df-en 7073  df-fin 7076  df-fi 7378  df-topgen 13626  df-ordt 13684  df-ps 14588  df-tsr 14589  df-top 16922  df-bases 16924  df-topon 16925  df-haus 17337
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