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Theorem t1sep 20041
Description: Any two distinct points in a T1 space are separated by an open set. (Contributed by Jeff Hankins, 1-Feb-2010.)
Hypothesis
Ref Expression
t1sep.1  |-  X  = 
U. J
Assertion
Ref Expression
t1sep  |-  ( ( J  e.  Fre  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B
) )  ->  E. o  e.  J  ( A  e.  o  /\  -.  B  e.  o ) )
Distinct variable groups:    A, o    B, o    o, J    o, X

Proof of Theorem t1sep
StepHypRef Expression
1 simpr3 1002 . . 3  |-  ( ( J  e.  Fre  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B
) )  ->  A  =/=  B )
2 t1sep.1 . . . . . 6  |-  X  = 
U. J
32t1sep2 20040 . . . . 5  |-  ( ( J  e.  Fre  /\  A  e.  X  /\  B  e.  X )  ->  ( A. o  e.  J  ( A  e.  o  ->  B  e.  o )  ->  A  =  B ) )
433adant3r3 1205 . . . 4  |-  ( ( J  e.  Fre  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B
) )  ->  ( A. o  e.  J  ( A  e.  o  ->  B  e.  o )  ->  A  =  B ) )
54necon3ad 2664 . . 3  |-  ( ( J  e.  Fre  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B
) )  ->  ( A  =/=  B  ->  -.  A. o  e.  J  ( A  e.  o  ->  B  e.  o )
) )
61, 5mpd 15 . 2  |-  ( ( J  e.  Fre  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B
) )  ->  -.  A. o  e.  J  ( A  e.  o  ->  B  e.  o )
)
7 rexanali 2907 . 2  |-  ( E. o  e.  J  ( A  e.  o  /\  -.  B  e.  o
)  <->  -.  A. o  e.  J  ( A  e.  o  ->  B  e.  o ) )
86, 7sylibr 212 1  |-  ( ( J  e.  Fre  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B
) )  ->  E. o  e.  J  ( A  e.  o  /\  -.  B  e.  o ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804   E.wrex 2805   U.cuni 4235   Frect1 19978
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-topgen 14936  df-top 19569  df-topon 19572  df-cld 19690  df-t1 19985
This theorem is referenced by: (None)
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