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Theorem t1sep 19107
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 996 . . 3  |-  ( ( J  e.  Fre  /\  ( A  e.  X  /\  B  e.  X  /\  A  =/=  B
) )  ->  A  =/=  B )
2 t1sep.1 . . . . . 6  |-  X  = 
U. J
32t1sep2 19106 . . . . 5  |-  ( ( J  e.  Fre  /\  A  e.  X  /\  B  e.  X )  ->  ( A. o  e.  J  ( A  e.  o  ->  B  e.  o )  ->  A  =  B ) )
433adant3r3 1199 . . . 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 2662 . . 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 2883 . 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800   U.cuni 4200   Frect1 19044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-iota 5490  df-fun 5529  df-fv 5535  df-topgen 14502  df-top 18636  df-topon 18639  df-cld 18756  df-t1 19051
This theorem is referenced by: (None)
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