Theorem t1t0 21950

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 21932	. . 3 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
2		toptopon2 21520	. . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
3	1, 2	sylib 220	. 2 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘∪ 𝐽))
4		biimp 217	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜))
5	4	ralimi 3160	. . . . . . 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 3160	. . . . 5 ⊢ (∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) → ∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))
8	7	ralimi 3160	. . . 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 21949	. . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Fre ↔ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 → 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
11		ist0-2 21946	. . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Kol2 ↔ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)))
12	9, 10, 11	3imtr4d 296	. 2 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → (𝐽 ∈ Fre → 𝐽 ∈ Kol2))
13	3, 12	mpcom 38	1 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Kol2)

Syntax hints:   → wi 4   ↔ wb 208   ∈ wcel 2110  ∀wral 3138  ∪ cuni 4831  ‘cfv 6349  Topctop 21495  TopOnctopon 21512  Kol2ct0 21908  Frect1 21909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-topgen 16711  df-top 21496  df-topon 21513  df-cld 21621  df-t0 21915  df-t1 21916

This theorem is referenced by:  t1r0  22423  ist1-5  22424  ishaus3  22425  reghaus  22427  nrmhaus  22428  tgpt0  22721