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Theorem t1sep 17434
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 965 . . 3  |-  ( ( J  e.  Fre  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B
) )  ->  A  =/=  B )
2 t1sep.1 . . . . . 6  |-  X  = 
U. J
32t1sep2 17433 . . . . 5  |-  ( ( J  e.  Fre  /\  A  e.  X  /\  B  e.  X )  ->  ( A. o  e.  J  ( A  e.  o  ->  B  e.  o )  ->  A  =  B ) )
433adant3r3 1164 . . . 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 2637 . . 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 2751 . 2  |-  ( E. o  e.  J  ( A  e.  o  /\  -.  B  e.  o
)  <->  -.  A. o  e.  J  ( A  e.  o  ->  B  e.  o ) )
86, 7sylibr 204 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 set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706   U.cuni 4015   Frect1 17371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-topgen 13667  df-top 16963  df-topon 16966  df-cld 17083  df-t1 17378
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