Theorem t0sep 23635

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 23631	. . 3 ⊢ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (∀𝑥 ∈ 𝐽 (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) → 𝑦 = 𝑧)))
3	2	simprbi 503	. 2 ⊢ (𝐽 ∈ Kol2 → ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 (∀𝑥 ∈ 𝐽 (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) → 𝑦 = 𝑧))
4		eleq1 2849	. . . . . . 7 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ 𝑥 ↔ 𝐴 ∈ 𝑥))
5	4	bibi1d 346	. . . . . 6 ⊢ (𝑦 = 𝐴 → ((𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) ↔ (𝐴 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥)))
6	5	ralbidv 3186	. . . . 5 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝐽 (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥)))
7		eqeq1 2765	. . . . 5 ⊢ (𝑦 = 𝐴 → (𝑦 = 𝑧 ↔ 𝐴 = 𝑧))
8	6, 7	imbi12d 347	. . . 4 ⊢ (𝑦 = 𝐴 → ((∀𝑥 ∈ 𝐽 (𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) → 𝑦 = 𝑧) ↔ (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) → 𝐴 = 𝑧)))
9		eleq1 2849	. . . . . . 7 ⊢ (𝑧 = 𝐵 → (𝑧 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥))
10	9	bibi2d 345	. . . . . 6 ⊢ (𝑧 = 𝐵 → ((𝐴 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) ↔ (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥)))
11	10	ralbidv 3186	. . . . 5 ⊢ (𝑧 = 𝐵 → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) ↔ ∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥)))
12		eqeq2 2773	. . . . 5 ⊢ (𝑧 = 𝐵 → (𝐴 = 𝑧 ↔ 𝐴 = 𝐵))
13	11, 12	imbi12d 347	. . . 4 ⊢ (𝑧 = 𝐵 → ((∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥) → 𝐴 = 𝑧) ↔ (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥) → 𝐴 = 𝐵)))
14	8, 13	rspc2va 3588	. . 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 464	. 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 592	1 ⊢ ((𝐽 ∈ Kol2 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑥 ∈ 𝐽 (𝐴 ∈ 𝑥 ↔ 𝐵 ∈ 𝑥) → 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∪ cuni 4867  Topctop 23204  Kol2ct0 23617

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-ss 3916  df-uni 4868  df-t0 23624

This theorem is used by:  t0dist  23636  cnt0  23657