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Theorem t1sep2 21221
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 21182 . . . . . 6 (𝐽 ∈ Fre → 𝐽 ∈ Top)
2 t1sep.1 . . . . . . 7 𝑋 = 𝐽
32toptopon 20770 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
41, 3sylib 208 . . . . 5 (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘𝑋))
5 ist1-2 21199 . . . . 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 256 . . 3 (𝐽 ∈ Fre → ∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦))
8 eleq1 2718 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑜𝐴𝑜))
98imbi1d 330 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝑜𝑦𝑜) ↔ (𝐴𝑜𝑦𝑜)))
109ralbidv 3015 . . . . 5 (𝑥 = 𝐴 → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜𝑦𝑜)))
11 eqeq1 2655 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
1210, 11imbi12d 333 . . . 4 (𝑥 = 𝐴 → ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (∀𝑜𝐽 (𝐴𝑜𝑦𝑜) → 𝐴 = 𝑦)))
13 eleq1 2718 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝑜𝐵𝑜))
1413imbi2d 329 . . . . . 6 (𝑦 = 𝐵 → ((𝐴𝑜𝑦𝑜) ↔ (𝐴𝑜𝐵𝑜)))
1514ralbidv 3015 . . . . 5 (𝑦 = 𝐵 → (∀𝑜𝐽 (𝐴𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))
16 eqeq2 2662 . . . . 5 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
1715, 16imbi12d 333 . . . 4 (𝑦 = 𝐵 → ((∀𝑜𝐽 (𝐴𝑜𝑦𝑜) → 𝐴 = 𝑦) ↔ (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵)))
1812, 17rspc2v 3353 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵)))
197, 18mpan9 485 . 2 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
20193impb 1279 1 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐵𝑋) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941   cuni 4468  cfv 5926  Topctop 20746  TopOnctopon 20763  Frect1 21159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-topgen 16151  df-top 20747  df-topon 20764  df-cld 20871  df-t1 21166
This theorem is referenced by:  t1sep  21222  isr0  21588
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