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Theorem t0sep 21176
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 21172 . . 3 (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑦𝑋𝑧𝑋 (∀𝑥𝐽 (𝑦𝑥𝑧𝑥) → 𝑦 = 𝑧)))
32simprbi 479 . 2 (𝐽 ∈ Kol2 → ∀𝑦𝑋𝑧𝑋 (∀𝑥𝐽 (𝑦𝑥𝑧𝑥) → 𝑦 = 𝑧))
4 eleq1 2718 . . . . . . 7 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
54bibi1d 332 . . . . . 6 (𝑦 = 𝐴 → ((𝑦𝑥𝑧𝑥) ↔ (𝐴𝑥𝑧𝑥)))
65ralbidv 3015 . . . . 5 (𝑦 = 𝐴 → (∀𝑥𝐽 (𝑦𝑥𝑧𝑥) ↔ ∀𝑥𝐽 (𝐴𝑥𝑧𝑥)))
7 eqeq1 2655 . . . . 5 (𝑦 = 𝐴 → (𝑦 = 𝑧𝐴 = 𝑧))
86, 7imbi12d 333 . . . 4 (𝑦 = 𝐴 → ((∀𝑥𝐽 (𝑦𝑥𝑧𝑥) → 𝑦 = 𝑧) ↔ (∀𝑥𝐽 (𝐴𝑥𝑧𝑥) → 𝐴 = 𝑧)))
9 eleq1 2718 . . . . . . 7 (𝑧 = 𝐵 → (𝑧𝑥𝐵𝑥))
109bibi2d 331 . . . . . 6 (𝑧 = 𝐵 → ((𝐴𝑥𝑧𝑥) ↔ (𝐴𝑥𝐵𝑥)))
1110ralbidv 3015 . . . . 5 (𝑧 = 𝐵 → (∀𝑥𝐽 (𝐴𝑥𝑧𝑥) ↔ ∀𝑥𝐽 (𝐴𝑥𝐵𝑥)))
12 eqeq2 2662 . . . . 5 (𝑧 = 𝐵 → (𝐴 = 𝑧𝐴 = 𝐵))
1311, 12imbi12d 333 . . . 4 (𝑧 = 𝐵 → ((∀𝑥𝐽 (𝐴𝑥𝑧𝑥) → 𝐴 = 𝑧) ↔ (∀𝑥𝐽 (𝐴𝑥𝐵𝑥) → 𝐴 = 𝐵)))
148, 13rspc2va 3354 . . 3 (((𝐴𝑋𝐵𝑋) ∧ ∀𝑦𝑋𝑧𝑋 (∀𝑥𝐽 (𝑦𝑥𝑧𝑥) → 𝑦 = 𝑧)) → (∀𝑥𝐽 (𝐴𝑥𝐵𝑥) → 𝐴 = 𝐵))
1514ancoms 468 . 2 ((∀𝑦𝑋𝑧𝑋 (∀𝑥𝐽 (𝑦𝑥𝑧𝑥) → 𝑦 = 𝑧) ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑥𝐽 (𝐴𝑥𝐵𝑥) → 𝐴 = 𝐵))
163, 15sylan 487 1 ((𝐽 ∈ Kol2 ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑥𝐽 (𝐴𝑥𝐵𝑥) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941   cuni 4468  Topctop 20746  Kol2ct0 21158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-uni 4469  df-t0 21165
This theorem is referenced by:  t0dist  21177  cnt0  21198
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