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Theorem hausnlly 22237
Description: A Hausdorff space is n-locally Hausdorff iff it is locally Hausdorff (both notions are thus referred to as "locally Hausdorff"). (Contributed by Mario Carneiro, 2-Mar-2015.)
Assertion
Ref Expression
hausnlly (𝐽 ∈ 𝑛-Locally Haus ↔ 𝐽 ∈ Locally Haus)

Proof of Theorem hausnlly
Dummy variables 𝑗 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 resthaus 22112 . . . . 5 ((𝑗 ∈ Haus ∧ 𝑥𝑗) → (𝑗t 𝑥) ∈ Haus)
21adantl 485 . . . 4 ((⊤ ∧ (𝑗 ∈ Haus ∧ 𝑥𝑗)) → (𝑗t 𝑥) ∈ Haus)
32restnlly 22226 . . 3 (⊤ → 𝑛-Locally Haus = Locally Haus)
43mptru 1549 . 2 𝑛-Locally Haus = Locally Haus
54eleq2i 2824 1 (𝐽 ∈ 𝑛-Locally Haus ↔ 𝐽 ∈ Locally Haus)
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1542  wtru 1543  wcel 2113  (class class class)co 7164  t crest 16790  Hauscha 22052  Locally clly 22208  𝑛-Locally cnlly 22209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-int 4834  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-map 8432  df-en 8549  df-fin 8552  df-fi 8941  df-rest 16792  df-topgen 16813  df-top 21638  df-topon 21655  df-bases 21690  df-nei 21842  df-cn 21971  df-haus 22059  df-lly 22210  df-nlly 22211
This theorem is referenced by: (None)
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