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Theorem gneispa 37448
 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 4280 . . . . . . 7 (𝑝𝑋 → {𝑝} ⊆ 𝑋)
2 gneispace.x . . . . . . . 8 𝑋 = 𝐽
32tpnei 20735 . . . . . . 7 (𝐽 ∈ Top → ({𝑝} ⊆ 𝑋𝑋 ∈ ((nei‘𝐽)‘{𝑝})))
41, 3syl5ib 233 . . . . . 6 (𝐽 ∈ Top → (𝑝𝑋𝑋 ∈ ((nei‘𝐽)‘{𝑝})))
54imp 444 . . . . 5 ((𝐽 ∈ Top ∧ 𝑝𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝑝}))
6 ne0i 3880 . . . . 5 (𝑋 ∈ ((nei‘𝐽)‘{𝑝}) → ((nei‘𝐽)‘{𝑝}) ≠ ∅)
75, 6syl 17 . . . 4 ((𝐽 ∈ Top ∧ 𝑝𝑋) → ((nei‘𝐽)‘{𝑝}) ≠ ∅)
8 elnei 20725 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑝𝑋𝑛 ∈ ((nei‘𝐽)‘{𝑝})) → 𝑝𝑛)
983expia 1259 . . . . 5 ((𝐽 ∈ Top ∧ 𝑝𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑝}) → 𝑝𝑛))
109ralrimiv 2948 . . . 4 ((𝐽 ∈ Top ∧ 𝑝𝑋) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛)
117, 10jca 553 . . 3 ((𝐽 ∈ Top ∧ 𝑝𝑋) → (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛))
1211ex 449 . 2 (𝐽 ∈ Top → (𝑝𝑋 → (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛)))
1312ralrimiv 2948 1 (𝐽 ∈ Top → ∀𝑝𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896   ⊆ wss 3540  ∅c0 3874  {csn 4125  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  neicnei 20711 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-nei 20712 This theorem is referenced by: (None)
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