Theorem t0sep 22475

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 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥) → 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.

Step	Hyp	Ref	Expression
1		ist0.1	. . . 4 ⊢ 𝑋 = ∪ 𝐽
2	1	ist0 22471	. . 3 ⊢ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (∀𝑥 ∈ 𝐽 (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) → 𝑦 = 𝑧)))
3	2	simprbi 497	. 2 ⊢ (𝐽 ∈ Kol2 → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (∀𝑥 ∈ 𝐽 (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) → 𝑦 = 𝑧))
4		eleq1 2826	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
5	4	bibi1d 344	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) ↔ (𝐴 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥)))
6	5	ralbidv 3112	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐽 (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥)))
7		eqeq1 2742	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝑧 ↔ 𝐴 = 𝑧))
8	6, 7	imbi12d 345	. . . 4 ⊢ (𝑦 = 𝐴 → ((∀𝑥 ∈ 𝐽 (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) → 𝑦 = 𝑧) ↔ (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) → 𝐴 = 𝑧)))
9		eleq1 2826	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥))
10	9	bibi2d 343	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) ↔ (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥)))
11	10	ralbidv 3112	. . . . 5 ⊢ (𝑧 = 𝐵 → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥)))
12		eqeq2 2750	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝐴 = 𝑧 ↔ 𝐴 = 𝐵))
13	11, 12	imbi12d 345	. . . 4 ⊢ (𝑧 = 𝐵 → ((∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) → 𝐴 = 𝑧) ↔ (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥) → 𝐴 = 𝐵)))
14	8, 13	rspc2va 3571	. . 3 ⊢ (((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (∀𝑥 ∈ 𝐽 (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) → 𝑦 = 𝑧)) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥) → 𝐴 = 𝐵))
15	14	ancoms 459	. 2 ⊢ ((∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (∀𝑥 ∈ 𝐽 (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) → 𝑦 = 𝑧) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥) → 𝐴 = 𝐵))
16	3, 15	sylan 580	1 ⊢ ((𝐽 ∈ Kol2 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥) → 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∪ cuni 4839  Topctop 22042  Kol2ct0 22457

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-t0 22464

This theorem is referenced by:  t0dist  22476  cnt0  22497