Theorem t1t0 20962

Description: A T₁ space is a T₀ space. (Contributed by Jeff Hankins, 1-Feb-2010.)

Assertion

Ref	Expression
t1t0	⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2)

Proof of Theorem t1t0

Dummy variables 𝑥 𝑦 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		t1top 20944	. . 3 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
2		eqid 2610	. . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽
3	2	toptopon 20548	. . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
4	1, 3	sylib 207	. 2 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘∪ 𝐽))
5		biimp 204	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
6	5	ralimi 2936	. . . . . . 7 ⊢ (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
7	6	imim1i 61	. . . . . 6 ⊢ ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
8	7	ralimi 2936	. . . . 5 ⊢ (∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) → ∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
9	8	ralimi 2936	. . . 4 ⊢ (∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) → ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
10	9	a1i 11	. . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) → ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
11		ist1-2 20961	. . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
12		ist0-2 20958	. . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
13	10, 11, 12	3imtr4d 282	. 2 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Fre → 𝐽 ∈ Kol2))
14	4, 13	mpcom 37	1 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2)

Syntax hints:   → wi 4   ↔ wb 195   ∈ wcel 1977  ∀wral 2896  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  TopOnctopon 20518  Kol2ct0 20920  Frect1 20921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-topgen 15927  df-top 20521  df-topon 20523  df-cld 20633  df-t0 20927  df-t1 20928

This theorem is referenced by:  t1r0  21434  ist1-5  21435  ishaus3  21436  reghaus  21438  nrmhaus  21439  tgpt0  21732