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Theorem ordthaus 17329
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 2366 . . . . . 6  |-  dom  R  =  dom  R
21ordthauslem 17328 . . . . 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 17328 . . . . . . 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 2610 . . . . . . . 8  |-  ( y  =/=  x  <->  x  =/=  y )
5 3ancoma 942 . . . . . . . . . . 11  |-  ( ( y  e.  n  /\  x  e.  m  /\  ( n  i^i  m
)  =  (/) )  <->  ( x  e.  m  /\  y  e.  n  /\  (
n  i^i  m )  =  (/) ) )
6 incom 3449 . . . . . . . . . . . . 13  |-  ( n  i^i  m )  =  ( m  i^i  n
)
76eqeq1i 2373 . . . . . . . . . . . 12  |-  ( ( n  i^i  m )  =  (/)  <->  ( m  i^i  n )  =  (/) )
873anbi3i 1145 . . . . . . . . . . 11  |-  ( ( x  e.  m  /\  y  e.  n  /\  ( n  i^i  m
)  =  (/) )  <->  ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) )
95, 8bitri 240 . . . . . . . . . 10  |-  ( ( y  e.  n  /\  x  e.  m  /\  ( n  i^i  m
)  =  (/) )  <->  ( x  e.  m  /\  y  e.  n  /\  (
m  i^i  n )  =  (/) ) )
1092rexbii 2655 . . . . . . . . 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 2786 . . . . . . . . 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 240 . . . . . . . 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 316 . . . . . . 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 217 . . . . . 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 1158 . . . . 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 14538 . . . . 5  |-  ( ( R  e.  TosetRel  /\  x  e.  dom  R  /\  y  e.  dom  R )  -> 
( x R y  \/  y R x ) )
172, 15, 16mpjaod 370 . . . 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 1153 . . 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 2720 . 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 17140 . . 3  |-  ( R  e.  TosetRel  ->  (ordTop `  R )  e.  (TopOn `  dom  R ) )
21 ishaus2 17296 . . 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 15 . 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 223 1  |-  ( R  e.  TosetRel  ->  (ordTop `  R )  e.  Haus )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ w3a 935    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   E.wrex 2629    i^i cin 3237   (/)c0 3543   class class class wbr 4125   dom cdm 4792   ` cfv 5358  ordTopcordt 13608    TosetRel ctsr 14512  TopOnctopon 16849   Hauscha 17253
This theorem is referenced by:  xrge0tsms  18553  xrhaus  23649  xrge0tsmsd  23735  xrge0haus  23806
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-er 6802  df-en 7007  df-fin 7010  df-fi 7312  df-topgen 13554  df-ordt 13612  df-ps 14516  df-tsr 14517  df-top 16853  df-bases 16855  df-topon 16856  df-haus 17260
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