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Theorem t1sep 22744
Description: Any two distinct points in a T1 space are separated by an open set. (Contributed by Jeff Hankins, 1-Feb-2010.)
Hypothesis
Ref Expression
t1sep.1 𝑋 = 𝐽
Assertion
Ref Expression
t1sep ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ∃𝑜𝐽 (𝐴𝑜 ∧ ¬ 𝐵𝑜))
Distinct variable groups:   𝐴,𝑜   𝐵,𝑜   𝑜,𝐽   𝑜,𝑋

Proof of Theorem t1sep
StepHypRef Expression
1 simpr3 1197 . . 3 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → 𝐴𝐵)
2 t1sep.1 . . . . . 6 𝑋 = 𝐽
32t1sep2 22743 . . . . 5 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐵𝑋) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
433adant3r3 1185 . . . 4 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
54necon3ad 2953 . . 3 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → (𝐴𝐵 → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))
61, 5mpd 15 . 2 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜))
7 rexanali 3102 . 2 (∃𝑜𝐽 (𝐴𝑜 ∧ ¬ 𝐵𝑜) ↔ ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜))
86, 7sylibr 233 1 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ∃𝑜𝐽 (𝐴𝑜 ∧ ¬ 𝐵𝑜))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070   cuni 4869  Frect1 22681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-topgen 17333  df-top 22266  df-topon 22283  df-cld 22393  df-t1 22688
This theorem is referenced by: (None)
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