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Theorem sst1 23233
Description: A topology finer than a T1 topology is T1. (Contributed by Mario Carneiro, 25-Aug-2015.)
Hypothesis
Ref Expression
t1sep.1 𝑋 = 𝐽
Assertion
Ref Expression
sst1 ((𝐽 ∈ Fre ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ Fre)

Proof of Theorem sst1
StepHypRef Expression
1 t1sep.1 . 2 𝑋 = 𝐽
2 t1top 23189 . 2 (𝐽 ∈ Fre → 𝐽 ∈ Top)
3 cnt1 23209 . 2 ((𝐽 ∈ Fre ∧ ( I ↾ 𝑋):𝑋1-1𝑋 ∧ ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ Fre)
41, 2, 3sshauslem 23231 1 ((𝐽 ∈ Fre ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽𝐾) → 𝐾 ∈ Fre)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1533  wcel 2098  wss 3943   cuni 4902   I cid 5566  cres 5671  cfv 6537  TopOnctopon 22767  Frect1 23166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8824  df-top 22751  df-topon 22768  df-cld 22878  df-cn 23086  df-t1 23173
This theorem is referenced by: (None)
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