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Theorem t0dist 21861
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 21860 . . . . 5 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
32necon3ad 3026 . . . 4 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐵 → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))
43exp32 421 . . 3 (𝐽 ∈ Kol2 → (𝐴𝑋 → (𝐵𝑋 → (𝐴𝐵 → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))))
543imp2 1341 . 2 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜))
6 rexnal 3235 . 2 (∃𝑜𝐽 ¬ (𝐴𝑜𝐵𝑜) ↔ ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜))
75, 6sylibr 235 1 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ∃𝑜𝐽 ¬ (𝐴𝑜𝐵𝑜))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136   cuni 4830  Kol2ct0 21842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-uni 4831  df-t0 21849
This theorem is referenced by: (None)
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