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Theorem ordthaus 19679
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 2467 . . . . . 6  |-  dom  R  =  dom  R
21ordthauslem 19678 . . . . 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 19678 . . . . . . 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 2736 . . . . . . . 8  |-  ( y  =/=  x  <->  x  =/=  y )
5 3ancoma 980 . . . . . . . . . . 11  |-  ( ( y  e.  n  /\  x  e.  m  /\  ( n  i^i  m
)  =  (/) )  <->  ( x  e.  m  /\  y  e.  n  /\  (
n  i^i  m )  =  (/) ) )
6 incom 3691 . . . . . . . . . . . . 13  |-  ( n  i^i  m )  =  ( m  i^i  n
)
76eqeq1i 2474 . . . . . . . . . . . 12  |-  ( ( n  i^i  m )  =  (/)  <->  ( m  i^i  n )  =  (/) )
873anbi3i 1189 . . . . . . . . . . 11  |-  ( ( x  e.  m  /\  y  e.  n  /\  ( n  i^i  m
)  =  (/) )  <->  ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) )
95, 8bitri 249 . . . . . . . . . 10  |-  ( ( y  e.  n  /\  x  e.  m  /\  ( n  i^i  m
)  =  (/) )  <->  ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) )
1092rexbii 2966 . . . . . . . . 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 3023 . . . . . . . . 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 249 . . . . . . . 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 326 . . . . . . 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 226 . . . . . 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 1202 . . . . 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 15706 . . . . 5  |-  ( ( R  e.  TosetRel  /\  x  e.  dom  R  /\  y  e.  dom  R )  -> 
( x R y  \/  y R x ) )
172, 15, 16mpjaod 381 . . . 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 1197 . . 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 2885 . 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 19488 . . 3  |-  ( R  e.  TosetRel  ->  (ordTop `  R )  e.  (TopOn `  dom  R ) )
21 ishaus2 19646 . . 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 232 1  |-  ( R  e.  TosetRel  ->  (ordTop `  R )  e.  Haus )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    i^i cin 3475   (/)c0 3785   class class class wbr 4447   dom cdm 4999   ` cfv 5588  ordTopcordt 14754    TosetRel ctsr 15686  TopOnctopon 19190   Hauscha 19603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-1o 7130  df-oadd 7134  df-er 7311  df-en 7517  df-fin 7520  df-fi 7871  df-topgen 14699  df-ordt 14756  df-ps 15687  df-tsr 15688  df-top 19194  df-bases 19196  df-topon 19197  df-haus 19610
This theorem is referenced by:  xrge0tsms  21102  xrhaus  27280  xrge0tsmsd  27466
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