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Theorem t1sep 21973
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 1193 . . 3 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → 𝐴𝐵)
2 t1sep.1 . . . . . 6 𝑋 = 𝐽
32t1sep2 21972 . . . . 5 ((𝐽 ∈ Fre ∧ 𝐴𝑋𝐵𝑋) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
433adant3r3 1181 . . . 4 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → (∀𝑜𝐽 (𝐴𝑜𝐵𝑜) → 𝐴 = 𝐵))
54necon3ad 3024 . . 3 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → (𝐴𝐵 → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜)))
61, 5mpd 15 . 2 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜))
7 rexanali 3251 . 2 (∃𝑜𝐽 (𝐴𝑜 ∧ ¬ 𝐵𝑜) ↔ ¬ ∀𝑜𝐽 (𝐴𝑜𝐵𝑜))
86, 7sylibr 237 1 ((𝐽 ∈ Fre ∧ (𝐴𝑋𝐵𝑋𝐴𝐵)) → ∃𝑜𝐽 (𝐴𝑜 ∧ ¬ 𝐵𝑜))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  wne 3011  wral 3130  wrex 3131   cuni 4813  Frect1 21910
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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
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 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-topgen 16708  df-top 21497  df-topon 21514  df-cld 21622  df-t1 21917
This theorem is referenced by: (None)
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