Theorem t1t0 22851

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 22833	. . 3 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
2		toptopon2 22419	. . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 217	. 2 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		biimp 214	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
5	4	ralimi 3083	. . . . . . 7 ⊢ (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → ∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
6	5	imim1i 63	. . . . . 6 ⊢ ((∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) → (∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
7	6	ralimi 3083	. . . . 5 ⊢ (∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) → ∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
8	7	ralimi 3083	. . . 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))
9	8	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)))
10		ist1-2 22850	. . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
11		ist0-2 22847	. . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
12	9, 10, 11	3imtr4d 293	. 2 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Fre → 𝐽 ∈ Kol2))
13	3, 12	mpcom 38	1 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2)

Syntax hints:   → wi 4   ↔ wb 205   ∈ wcel 2106  ∀wral 3061  ∪ cuni 4908  ‘cfv 6543  Topctop 22394  TopOnctopon 22411  Kol2ct0 22809  Frect1 22810

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-topgen 17388  df-top 22395  df-topon 22412  df-cld 22522  df-t0 22816  df-t1 22817

This theorem is referenced by:  t1r0  23324  ist1-5  23325  ishaus3  23326  reghaus  23328  nrmhaus  23329  tgpt0  23622