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Theorem t0dist 20939
 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 20938 . . . . 5 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
32necon3ad 2795 . . . 4 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐵 → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))
43exp32 629 . . 3 (𝐽 ∈ Kol2 → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝐵 → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))))
543imp2 1274 . 2 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜))
6 rexnal 2978 . 2 (∃𝑜𝐽 ¬ (𝐴𝑜𝐵𝑜) ↔ ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜))
75, 6sylibr 223 1 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ∃𝑜𝐽 ¬ (𝐴𝑜𝐵𝑜))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∪ cuni 4372  Kol2ct0 20920 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-uni 4373  df-t0 20927 This theorem is referenced by: (None)
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