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Theorem ist1-3 22845
Description: A space is T1 iff every point is the only point in the intersection of all open sets containing that point. (Contributed by Jeff Hankins, 31-Jan-2010.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
Assertion
Ref Expression
ist1-3 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Fre ↔ βˆ€π‘₯ ∈ 𝑋 ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯}))
Distinct variable groups:   π‘₯,π‘œ,𝐽   π‘œ,𝑋,π‘₯
Proof of Theorem ist1-3
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ist1-2 22843 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Fre ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)))
2 toponmax 22420 . . . . . . . 8 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
3 eleq2 2823 . . . . . . . . 9 (π‘œ = 𝑋 β†’ (π‘₯ ∈ π‘œ ↔ π‘₯ ∈ 𝑋))
43intminss 4978 . . . . . . . 8 ((𝑋 ∈ 𝐽 ∧ π‘₯ ∈ 𝑋) β†’ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} βŠ† 𝑋)
52, 4sylan 581 . . . . . . 7 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} βŠ† 𝑋)
65sselda 3982 . . . . . 6 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}) β†’ 𝑦 ∈ 𝑋)
7 biimt 361 . . . . . 6 (𝑦 ∈ 𝑋 β†’ (𝑦 ∈ {π‘₯} ↔ (𝑦 ∈ 𝑋 β†’ 𝑦 ∈ {π‘₯})))
86, 7syl 17 . . . . 5 (((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) ∧ 𝑦 ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}) β†’ (𝑦 ∈ {π‘₯} ↔ (𝑦 ∈ 𝑋 β†’ 𝑦 ∈ {π‘₯})))
98ralbidva 3176 . . . 4 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ (βˆ€π‘¦ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}𝑦 ∈ {π‘₯} ↔ βˆ€π‘¦ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} (𝑦 ∈ 𝑋 β†’ 𝑦 ∈ {π‘₯})))
10 id 22 . . . . . . . . 9 (π‘₯ ∈ π‘œ β†’ π‘₯ ∈ π‘œ)
1110rgenw 3066 . . . . . . . 8 βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ π‘₯ ∈ π‘œ)
12 vex 3479 . . . . . . . . 9 π‘₯ ∈ V
1312elintrab 4964 . . . . . . . 8 (π‘₯ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} ↔ βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ π‘₯ ∈ π‘œ))
1411, 13mpbir 230 . . . . . . 7 π‘₯ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}
15 snssi 4811 . . . . . . 7 (π‘₯ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} β†’ {π‘₯} βŠ† ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ})
1614, 15ax-mp 5 . . . . . 6 {π‘₯} βŠ† ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}
17 eqss 3997 . . . . . 6 (∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯} ↔ (∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} βŠ† {π‘₯} ∧ {π‘₯} βŠ† ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}))
1816, 17mpbiran2 709 . . . . 5 (∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯} ↔ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} βŠ† {π‘₯})
19 dfss3 3970 . . . . 5 (∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} βŠ† {π‘₯} ↔ βˆ€π‘¦ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}𝑦 ∈ {π‘₯})
2018, 19bitri 275 . . . 4 (∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯} ↔ βˆ€π‘¦ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ}𝑦 ∈ {π‘₯})
21 vex 3479 . . . . . . . 8 𝑦 ∈ V
2221elintrab 4964 . . . . . . 7 (𝑦 ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} ↔ βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ))
23 velsn 4644 . . . . . . . 8 (𝑦 ∈ {π‘₯} ↔ 𝑦 = π‘₯)
24 equcom 2022 . . . . . . . 8 (𝑦 = π‘₯ ↔ π‘₯ = 𝑦)
2523, 24bitri 275 . . . . . . 7 (𝑦 ∈ {π‘₯} ↔ π‘₯ = 𝑦)
2622, 25imbi12i 351 . . . . . 6 ((𝑦 ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} β†’ 𝑦 ∈ {π‘₯}) ↔ (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦))
2726ralbii 3094 . . . . 5 (βˆ€π‘¦ ∈ 𝑋 (𝑦 ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} β†’ 𝑦 ∈ {π‘₯}) ↔ βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦))
28 ralcom3 3098 . . . . 5 (βˆ€π‘¦ ∈ 𝑋 (𝑦 ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} β†’ 𝑦 ∈ {π‘₯}) ↔ βˆ€π‘¦ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} (𝑦 ∈ 𝑋 β†’ 𝑦 ∈ {π‘₯}))
2927, 28bitr3i 277 . . . 4 (βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦) ↔ βˆ€π‘¦ ∈ ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} (𝑦 ∈ 𝑋 β†’ 𝑦 ∈ {π‘₯}))
309, 20, 293bitr4g 314 . . 3 ((𝐽 ∈ (TopOnβ€˜π‘‹) ∧ π‘₯ ∈ 𝑋) β†’ (∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯} ↔ βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)))
3130ralbidva 3176 . 2 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (βˆ€π‘₯ ∈ 𝑋 ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯} ↔ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (βˆ€π‘œ ∈ 𝐽 (π‘₯ ∈ π‘œ β†’ 𝑦 ∈ π‘œ) β†’ π‘₯ = 𝑦)))
321, 31bitr4d 282 1 (𝐽 ∈ (TopOnβ€˜π‘‹) β†’ (𝐽 ∈ Fre ↔ βˆ€π‘₯ ∈ 𝑋 ∩ {π‘œ ∈ 𝐽 ∣ π‘₯ ∈ π‘œ} = {π‘₯}))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  {crab 3433   βŠ† wss 3948  {csn 4628  βˆ© cint 4950  β€˜cfv 6541  TopOnctopon 22404  Frect1 22803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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-int 4951  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 6493  df-fun 6543  df-fv 6549  df-topgen 17386  df-top 22388  df-topon 22405  df-cld 22515  df-t1 22810
This theorem is referenced by: (None)
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