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Theorem t1sep2 21681
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 21642 . . . . . 6 (𝐽 ∈ Fre → 𝐽 ∈ Top)
2 t1sep.1 . . . . . . 7 𝑋 = 𝐽
32toptopon 21229 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
41, 3sylib 210 . . . . 5 (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘𝑋))
5 ist1-2 21659 . . . . 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 259 . . 3 (𝐽 ∈ Fre → ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
8 eleq1 2854 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑜𝐴𝑜))
98imbi1d 334 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝑜𝑦𝑜) ↔ (𝐴𝑜𝑦𝑜)))
109ralbidv 3148 . . . . 5 (𝑥 = 𝐴 → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜𝑦𝑜)))
11 eqeq1 2783 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
1210, 11imbi12d 337 . . . 4 (𝑥 = 𝐴 → ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (∀𝑜𝐽 (𝐴𝑜𝑦𝑜) → 𝐴 = 𝑦)))
13 eleq1 2854 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝑜𝐵𝑜))
1413imbi2d 333 . . . . . 6 (𝑦 = 𝐵 → ((𝐴𝑜𝑦𝑜) ↔ (𝐴𝑜𝐵𝑜)))
1514ralbidv 3148 . . . . 5 (𝑦 = 𝐵 → (∀𝑜𝐽 (𝐴𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))
16 eqeq2 2790 . . . . 5 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
1715, 16imbi12d 337 . . . 4 (𝑦 = 𝐵 → ((∀𝑜𝐽 (𝐴𝑜𝑦𝑜) → 𝐴 = 𝑦) ↔ (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵)))
1812, 17rspc2v 3549 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵)))
197, 18mpan9 499 . 2 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
20193impb 1095 1 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐵𝑋) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wral 3089   cuni 4712  cfv 6188  Topctop 21205  TopOnctopon 21222  Frect1 21619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-iota 6152  df-fun 6190  df-fv 6196  df-topgen 16573  df-top 21206  df-topon 21223  df-cld 21331  df-t1 21626
This theorem is referenced by:  t1sep  21682  isr0  22049
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