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Theorem t1sep 23378
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 = 𝐽
Assertion
Ref Expression
t1sep ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ∃𝑜𝐽 (𝐴𝑜 ∧ ¬ 𝐵𝑜))
Distinct variable groups:   𝐴,𝑜   𝐵,𝑜   𝑜,𝐽   𝑜,𝑋

Proof of Theorem t1sep
StepHypRef Expression
1 simpr3 1197 . . 3 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → 𝐴𝐵)
2 t1sep.1 . . . . . 6 𝑋 = 𝐽
32t1sep2 23377 . . . . 5 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐵𝑋) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
433adant3r3 1185 . . . 4 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
54necon3ad 2953 . . 3 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → (𝐴𝐵 → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))
61, 5mpd 15 . 2 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜))
7 rexanali 3102 . 2 (∃𝑜𝐽 (𝐴𝑜 ∧ ¬ 𝐵𝑜) ↔ ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜))
86, 7sylibr 234 1 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ∃𝑜𝐽 (𝐴𝑜 ∧ ¬ 𝐵𝑜))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070   cuni 4907  Frect1 23315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-topgen 17488  df-top 22900  df-topon 22917  df-cld 23027  df-t1 23322
This theorem is referenced by: (None)
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