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Theorem t0sep 21109
 Description: Any two topologically indistinguishable points in a T0 space are identical. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
ist0.1 𝑋 = 𝐽
Assertion
Ref Expression
t0sep ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑥𝐽 (𝐴𝑥𝐵𝑥) → 𝐴 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐽   𝑥,𝑋

Proof of Theorem t0sep
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ist0.1 . . . 4 𝑋 = 𝐽
21ist0 21105 . . 3 (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑧𝑋 (∀𝑥𝐽 (𝑦𝑥𝑧𝑥) → 𝑦 = 𝑧)))
32simprbi 480 . 2 (𝐽 ∈ Kol2 → ∀𝑦𝑋𝑧𝑋 (∀𝑥𝐽 (𝑦𝑥𝑧𝑥) → 𝑦 = 𝑧))
4 eleq1 2687 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
54bibi1d 333 . . . . . 6 (𝑦 = 𝐴 → ((𝑦𝑥𝑧𝑥) ↔ (𝐴𝑥𝑧𝑥)))
65ralbidv 2983 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝐽 (𝑦𝑥𝑧𝑥) ↔ ∀𝑥𝐽 (𝐴𝑥𝑧𝑥)))
7 eqeq1 2624 . . . . 5 (𝑦 = 𝐴 → (𝑦 = 𝑧𝐴 = 𝑧))
86, 7imbi12d 334 . . . 4 (𝑦 = 𝐴 → ((∀𝑥𝐽 (𝑦𝑥𝑧𝑥) → 𝑦 = 𝑧) ↔ (∀𝑥𝐽 (𝐴𝑥𝑧𝑥) → 𝐴 = 𝑧)))
9 eleq1 2687 . . . . . . 7 (𝑧 = 𝐵 → (𝑧𝑥𝐵𝑥))
109bibi2d 332 . . . . . 6 (𝑧 = 𝐵 → ((𝐴𝑥𝑧𝑥) ↔ (𝐴𝑥𝐵𝑥)))
1110ralbidv 2983 . . . . 5 (𝑧 = 𝐵 → (∀𝑥𝐽 (𝐴𝑥𝑧𝑥) ↔ ∀𝑥𝐽 (𝐴𝑥𝐵𝑥)))
12 eqeq2 2631 . . . . 5 (𝑧 = 𝐵 → (𝐴 = 𝑧𝐴 = 𝐵))
1311, 12imbi12d 334 . . . 4 (𝑧 = 𝐵 → ((∀𝑥𝐽 (𝐴𝑥𝑧𝑥) → 𝐴 = 𝑧) ↔ (∀𝑥𝐽 (𝐴𝑥𝐵𝑥) → 𝐴 = 𝐵)))
148, 13rspc2va 3318 . . 3 (((𝐴𝑋𝐵𝑋) ∧ ∀𝑦𝑋𝑧𝑋 (∀𝑥𝐽 (𝑦𝑥𝑧𝑥) → 𝑦 = 𝑧)) → (∀𝑥𝐽 (𝐴𝑥𝐵𝑥) → 𝐴 = 𝐵))
1514ancoms 469 . 2 ((∀𝑦𝑋𝑧𝑋 (∀𝑥𝐽 (𝑦𝑥𝑧𝑥) → 𝑦 = 𝑧) ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑥𝐽 (𝐴𝑥𝐵𝑥) → 𝐴 = 𝐵))
163, 15sylan 488 1 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑥𝐽 (𝐴𝑥𝐵𝑥) → 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1481   ∈ wcel 1988  ∀wral 2909  ∪ cuni 4427  Topctop 20679  Kol2ct0 21091 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-uni 4428  df-t0 21098 This theorem is referenced by:  t0dist  21110  cnt0  21131
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