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Theorem t1sep2 21977
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 21938 . . . . . 6 (𝐽 ∈ Fre → 𝐽 ∈ Top)
2 t1sep.1 . . . . . . 7 𝑋 = 𝐽
32toptopon 21525 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
41, 3sylib 221 . . . . 5 (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘𝑋))
5 ist1-2 21955 . . . . 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 270 . . 3 (𝐽 ∈ Fre → ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
8 eleq1 2880 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑜𝐴𝑜))
98imbi1d 345 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝑜𝑦𝑜) ↔ (𝐴𝑜𝑦𝑜)))
109ralbidv 3165 . . . . 5 (𝑥 = 𝐴 → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜𝑦𝑜)))
11 eqeq1 2805 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
1210, 11imbi12d 348 . . . 4 (𝑥 = 𝐴 → ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (∀𝑜𝐽 (𝐴𝑜𝑦𝑜) → 𝐴 = 𝑦)))
13 eleq1 2880 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝑜𝐵𝑜))
1413imbi2d 344 . . . . . 6 (𝑦 = 𝐵 → ((𝐴𝑜𝑦𝑜) ↔ (𝐴𝑜𝐵𝑜)))
1514ralbidv 3165 . . . . 5 (𝑦 = 𝐵 → (∀𝑜𝐽 (𝐴𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))
16 eqeq2 2813 . . . . 5 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
1715, 16imbi12d 348 . . . 4 (𝑦 = 𝐵 → ((∀𝑜𝐽 (𝐴𝑜𝑦𝑜) → 𝐴 = 𝑦) ↔ (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵)))
1812, 17rspc2v 3584 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵)))
197, 18mpan9 510 . 2 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
20193impb 1112 1 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐵𝑋) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wral 3109   cuni 4803  cfv 6328  Topctop 21501  TopOnctopon 21518  Frect1 21915
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 16712  df-top 21502  df-topon 21519  df-cld 21627  df-t1 21922
This theorem is referenced by:  t1sep  21978  isr0  22345
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