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Theorem t1t0 21957
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 21939 . . 3 (𝐽 ∈ Fre → 𝐽 ∈ Top)
2 toptopon2 21527 . . 3 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘ 𝐽))
31, 2sylib 221 . 2 (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘ 𝐽))
4 biimp 218 . . . . . . . 8 ((𝑥𝑜𝑦𝑜) → (𝑥𝑜𝑦𝑜))
54ralimi 3131 . . . . . . 7 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → ∀𝑜𝐽 (𝑥𝑜𝑦𝑜))
65imim1i 63 . . . . . 6 ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
76ralimi 3131 . . . . 5 (∀𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → ∀𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
87ralimi 3131 . . . 4 (∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
98a1i 11 . . 3 (𝐽 ∈ (TopOn‘ 𝐽) → (∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
10 ist1-2 21956 . . 3 (𝐽 ∈ (TopOn‘ 𝐽) → (𝐽 ∈ Fre ↔ ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
11 ist0-2 21953 . . 3 (𝐽 ∈ (TopOn‘ 𝐽) → (𝐽 ∈ Kol2 ↔ ∀𝑥 𝐽𝑦 𝐽(∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
129, 10, 113imtr4d 297 . 2 (𝐽 ∈ (TopOn‘ 𝐽) → (𝐽 ∈ Fre → 𝐽 ∈ Kol2))
133, 12mpcom 38 1 (𝐽 ∈ Fre → 𝐽 ∈ Kol2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wcel 2112  wral 3109   cuni 4803  cfv 6328  Topctop 21502  TopOnctopon 21519  Kol2ct0 21915  Frect1 21916
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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
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-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-topgen 16713  df-top 21503  df-topon 21520  df-cld 21628  df-t0 21922  df-t1 21923
This theorem is referenced by:  t1r0  22430  ist1-5  22431  ishaus3  22432  reghaus  22434  nrmhaus  22435  tgpt0  22728
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