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Theorem t1t0 21146
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.
StepHypRef Expression
1 t1top 21128 . . 3 (𝐽 ∈ Fre → 𝐽 ∈ Top)
2 eqid 2621 . . . 4 𝐽 = 𝐽
32toptopon 20716 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
41, 3sylib 208 . 2 (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘ 𝐽))
5 biimp 205 . . . . . . . 8 ((𝑥𝑜𝑦𝑜) → (𝑥𝑜𝑦𝑜))
65ralimi 2951 . . . . . . 7 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
76imim1i 63 . . . . . 6 ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
87ralimi 2951 . . . . 5 (∀𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → ∀𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
98ralimi 2951 . . . 4 (∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
109a1i 11 . . 3 (𝐽 ∈ (TopOn‘ 𝐽) → (∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
11 ist1-2 21145 . . 3 (𝐽 ∈ (TopOn‘ 𝐽) → (𝐽 ∈ Fre ↔ ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
12 ist0-2 21142 . . 3 (𝐽 ∈ (TopOn‘ 𝐽) → (𝐽 ∈ Kol2 ↔ ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
1310, 11, 123imtr4d 283 . 2 (𝐽 ∈ (TopOn‘ 𝐽) → (𝐽 ∈ Fre → 𝐽 ∈ Kol2))
144, 13mpcom 38 1 (𝐽 ∈ Fre → 𝐽 ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 1989  wral 2911   cuni 4434  cfv 5886  Topctop 20692  TopOnctopon 20709  Kol2ct0 21104  Frect1 21105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-iota 5849  df-fun 5888  df-fv 5894  df-topgen 16098  df-top 20693  df-topon 20710  df-cld 20817  df-t0 21111  df-t1 21112
This theorem is referenced by:  t1r0  21618  ist1-5  21619  ishaus3  21620  reghaus  21622  nrmhaus  21623  tgpt0  21916
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