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Theorem t0dist 23333
Description: Any two distinct points in a T0 space are topologically distinguishable. (Contributed by Jeff Hankins, 1-Feb-2010.)
Hypothesis
Ref Expression
ist0.1 𝑋 = 𝐽
Assertion
Ref Expression
t0dist ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ∃𝑜𝐽 ¬ (𝐴𝑜𝐵𝑜))
Distinct variable groups:   𝐴,𝑜   𝐵,𝑜   𝑜,𝐽   𝑜,𝑋

Proof of Theorem t0dist
StepHypRef Expression
1 ist0.1 . . . . . 6 𝑋 = 𝐽
21t0sep 23332 . . . . 5 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
32necon3ad 2953 . . . 4 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐵 → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))
43exp32 420 . . 3 (𝐽 ∈ Kol2 → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝐵 → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))))
543imp2 1350 . 2 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜))
6 rexnal 3100 . 2 (∃𝑜𝐽 ¬ (𝐴𝑜𝐵𝑜) ↔ ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜))
75, 6sylibr 234 1 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ∃𝑜𝐽 ¬ (𝐴𝑜𝐵𝑜))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070   cuni 4907  Kol2ct0 23314
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-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-ss 3968  df-uni 4908  df-t0 23321
This theorem is referenced by: (None)
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