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Theorem gneispa 37341
Description: Each point 𝑝 of the neighborhood space has at least one neighborhood; each neighborhood of 𝑝 contains 𝑝. Axiom A of Seifert And Threlfall. (Contributed by RP, 5-Apr-2021.)
Hypothesis
Ref Expression
gneispace.x 𝑋 = 𝐽
Assertion
Ref Expression
gneispa (𝐽 ∈ Top → ∀𝑝𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛))
Distinct variable groups:   𝑛,𝐽,𝑝   𝑛,𝑋
Allowed substitution hint:   𝑋(𝑝)

Proof of Theorem gneispa
StepHypRef Expression
1 snssi 4183 . . . . . . 7 (𝑝𝑋 → {𝑝} ⊆ 𝑋)
2 gneispace.x . . . . . . . 8 𝑋 = 𝐽
32tpnei 20659 . . . . . . 7 (𝐽 ∈ Top → ({𝑝} ⊆ 𝑋𝑋 ∈ ((nei‘𝐽)‘{𝑝})))
41, 3syl5ib 232 . . . . . 6 (𝐽 ∈ Top → (𝑝𝑋𝑋 ∈ ((nei‘𝐽)‘{𝑝})))
54imp 443 . . . . 5 ((𝐽 ∈ Top ∧ 𝑝𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝑝}))
6 ne0i 3783 . . . . 5 (𝑋 ∈ ((nei‘𝐽)‘{𝑝}) → ((nei‘𝐽)‘{𝑝}) ≠ ∅)
75, 6syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑝𝑋) → ((nei‘𝐽)‘{𝑝}) ≠ ∅)
8 elnei 20649 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑝𝑋𝑛 ∈ ((nei‘𝐽)‘{𝑝})) → 𝑝𝑛)
983expia 1258 . . . . 5 ((𝐽 ∈ Top ∧ 𝑝𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑝}) → 𝑝𝑛))
109ralrimiv 2852 . . . 4 ((𝐽 ∈ Top ∧ 𝑝𝑋) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛)
117, 10jca 552 . . 3 ((𝐽 ∈ Top ∧ 𝑝𝑋) → (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛))
1211ex 448 . 2 (𝐽 ∈ Top → (𝑝𝑋 → (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛)))
1312ralrimiv 2852 1 (𝐽 ∈ Top → ∀𝑝𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1938  wne 2684  wral 2800  wss 3444  c0 3777  {csn 4028   cuni 4270  cfv 5689  Topctop 20441  neicnei 20635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-top 20445  df-nei 20636
This theorem is referenced by: (None)
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