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Theorem hausnlly 22644
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 22519 . . . . 5 ((𝑗 ∈ Haus ∧ 𝑥𝑗) → (𝑗t 𝑥) ∈ Haus)
21adantl 482 . . . 4 ((⊤ ∧ (𝑗 ∈ Haus ∧ 𝑥𝑗)) → (𝑗t 𝑥) ∈ Haus)
32restnlly 22633 . . 3 (⊤ → 𝑛-Locally Haus = Locally Haus)
43mptru 1546 . 2 𝑛-Locally Haus = Locally Haus
54eleq2i 2830 1 (𝐽 ∈ 𝑛-Locally Haus ↔ 𝐽 ∈ Locally Haus)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396   = wceq 1539  wtru 1540  wcel 2106  (class class class)co 7275  t crest 17131  Hauscha 22459  Locally clly 22615  𝑛-Locally cnlly 22616
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-map 8617  df-en 8734  df-fin 8737  df-fi 9170  df-rest 17133  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-nei 22249  df-cn 22378  df-haus 22466  df-lly 22617  df-nlly 22618
This theorem is referenced by: (None)
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