Theorem t0dist 21930

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 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ≠ 𝐵)) → ∃𝑜 ∈ 𝐽 ¬ (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜))

Distinct variable groups:   𝐴,𝑜   𝐵,𝑜   𝑜,𝐽   𝑜,𝑋

Proof of Theorem t0dist

Step	Hyp	Ref	Expression
1		ist0.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
2	1	t0sep 21929	. . . . 5 ⊢ ((𝐽 ∈ Kol2 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜) → 𝐴 = 𝐵))
3	2	necon3ad 3000	. . . 4 ⊢ ((𝐽 ∈ Kol2 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴 ≠ 𝐵 → ¬ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜)))
4	3	exp32 424	. . 3 ⊢ (𝐽 ∈ Kol2 → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → (𝐴 ≠ 𝐵 → ¬ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜)))))
5	4	3imp2 1346	. 2 ⊢ ((𝐽 ∈ Kol2 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ≠ 𝐵)) → ¬ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜))
6		rexnal 3201	. 2 ⊢ (∃𝑜 ∈ 𝐽 ¬ (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜) ↔ ¬ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜))
7	5, 6	sylibr 237	1 ⊢ ((𝐽 ∈ Kol2 ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ≠ 𝐵)) → ∃𝑜 ∈ 𝐽 ¬ (𝐴 ∈ 𝑜 ↔ 𝐵 ∈ 𝑜))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∪ cuni 4800  Kol2ct0 21911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-in 3888  df-ss 3898  df-uni 4801  df-t0 21918

This theorem is referenced by: (None)