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Theorem t1sep2 23193
Description: Any two points in a T1 space which have no separation are equal. (Contributed by Jeff Hankins, 1-Feb-2010.)
Hypothesis
Ref Expression
t1sep.1 𝑋 = 𝐽
Assertion
Ref Expression
t1sep2 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐵𝑋) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
Distinct variable groups:   𝐴,𝑜   𝐵,𝑜   𝑜,𝐽   𝑜,𝑋

Proof of Theorem t1sep2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 t1top 23154 . . . . . 6 (𝐽 ∈ Fre → 𝐽 ∈ Top)
2 t1sep.1 . . . . . . 7 𝑋 = 𝐽
32toptopon 22739 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
41, 3sylib 217 . . . . 5 (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘𝑋))
5 ist1-2 23171 . . . . 5 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
64, 5syl 17 . . . 4 (𝐽 ∈ Fre → (𝐽 ∈ Fre ↔ ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)))
76ibi 267 . . 3 (𝐽 ∈ Fre → ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
8 eleq1 2820 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑜𝐴𝑜))
98imbi1d 341 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝑜𝑦𝑜) ↔ (𝐴𝑜𝑦𝑜)))
109ralbidv 3176 . . . . 5 (𝑥 = 𝐴 → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜𝑦𝑜)))
11 eqeq1 2735 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
1210, 11imbi12d 344 . . . 4 (𝑥 = 𝐴 → ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (∀𝑜𝐽 (𝐴𝑜𝑦𝑜) → 𝐴 = 𝑦)))
13 eleq1 2820 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝑜𝐵𝑜))
1413imbi2d 340 . . . . . 6 (𝑦 = 𝐵 → ((𝐴𝑜𝑦𝑜) ↔ (𝐴𝑜𝐵𝑜)))
1514ralbidv 3176 . . . . 5 (𝑦 = 𝐵 → (∀𝑜𝐽 (𝐴𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))
16 eqeq2 2743 . . . . 5 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
1715, 16imbi12d 344 . . . 4 (𝑦 = 𝐵 → ((∀𝑜𝐽 (𝐴𝑜𝑦𝑜) → 𝐴 = 𝑦) ↔ (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵)))
1812, 17rspc2v 3622 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵)))
197, 18mpan9 506 . 2 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
20193impb 1114 1 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐵𝑋) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060   cuni 4908  cfv 6543  Topctop 22715  TopOnctopon 22732  Frect1 23131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  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-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 6495  df-fun 6545  df-fv 6551  df-topgen 17396  df-top 22716  df-topon 22733  df-cld 22843  df-t1 23138
This theorem is referenced by:  t1sep  23194  isr0  23561
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