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Theorem t1sep2 21154
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 21115 . . . . . 6 (𝐽 ∈ Fre → 𝐽 ∈ Top)
2 t1sep.1 . . . . . . 7 𝑋 = 𝐽
32toptopon 20703 . . . . . 6 (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
41, 3sylib 208 . . . . 5 (𝐽 ∈ Fre → 𝐽 ∈ (TopOn‘𝑋))
5 ist1-2 21132 . . . . 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 2687 . . . . . . 7 (𝑥 = 𝐴 → (𝑥𝑜𝐴𝑜))
98imbi1d 331 . . . . . 6 (𝑥 = 𝐴 → ((𝑥𝑜𝑦𝑜) ↔ (𝐴𝑜𝑦𝑜)))
109ralbidv 2983 . . . . 5 (𝑥 = 𝐴 → (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜𝑦𝑜)))
11 eqeq1 2624 . . . . 5 (𝑥 = 𝐴 → (𝑥 = 𝑦𝐴 = 𝑦))
1210, 11imbi12d 334 . . . 4 (𝑥 = 𝐴 → ((∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) ↔ (∀𝑜𝐽 (𝐴𝑜𝑦𝑜) → 𝐴 = 𝑦)))
13 eleq1 2687 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝑜𝐵𝑜))
1413imbi2d 330 . . . . . 6 (𝑦 = 𝐵 → ((𝐴𝑜𝑦𝑜) ↔ (𝐴𝑜𝐵𝑜)))
1514ralbidv 2983 . . . . 5 (𝑦 = 𝐵 → (∀𝑜𝐽 (𝐴𝑜𝑦𝑜) ↔ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))
16 eqeq2 2631 . . . . 5 (𝑦 = 𝐵 → (𝐴 = 𝑦𝐴 = 𝐵))
1715, 16imbi12d 334 . . . 4 (𝑦 = 𝐵 → ((∀𝑜𝐽 (𝐴𝑜𝑦𝑜) → 𝐴 = 𝑦) ↔ (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵)))
1812, 17rspc2v 3317 . . 3 ((𝐴𝑋𝐵𝑋) → (∀𝑥𝑋𝑦𝑋 (∀𝑜𝐽 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵)))
197, 18mpan9 486 . 2 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
20193impb 1258 1 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐵𝑋) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909   cuni 4427  cfv 5876  Topctop 20679  TopOnctopon 20696  Frect1 21092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-topgen 16085  df-top 20680  df-topon 20697  df-cld 20804  df-t1 21099
This theorem is referenced by:  t1sep  21155  isr0  21521
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