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Theorem gneispa 41740
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 4741 . . . . . 6 (𝑝𝑋 → {𝑝} ⊆ 𝑋)
2 gneispace.x . . . . . . 7 𝑋 = 𝐽
32tpnei 22272 . . . . . 6 (𝐽 ∈ Top → ({𝑝} ⊆ 𝑋𝑋 ∈ ((nei‘𝐽)‘{𝑝})))
41, 3syl5ib 243 . . . . 5 (𝐽 ∈ Top → (𝑝𝑋𝑋 ∈ ((nei‘𝐽)‘{𝑝})))
54imp 407 . . . 4 ((𝐽 ∈ Top ∧ 𝑝𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝑝}))
65ne0d 4269 . . 3 ((𝐽 ∈ Top ∧ 𝑝𝑋) → ((nei‘𝐽)‘{𝑝}) ≠ ∅)
7 elnei 22262 . . . . 5 ((𝐽 ∈ Top ∧ 𝑝𝑋𝑛 ∈ ((nei‘𝐽)‘{𝑝})) → 𝑝𝑛)
873expia 1120 . . . 4 ((𝐽 ∈ Top ∧ 𝑝𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑝}) → 𝑝𝑛))
98ralrimiv 3102 . . 3 ((𝐽 ∈ Top ∧ 𝑝𝑋) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛)
106, 9jca 512 . 2 ((𝐽 ∈ Top ∧ 𝑝𝑋) → (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛))
1110ralrimiva 3103 1 (𝐽 ∈ Top → ∀𝑝𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wcel 2106  wne 2943  wral 3064  wss 3887  c0 4256  {csn 4561   cuni 4839  cfv 6433  Topctop 22042  neicnei 22248
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
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  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-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  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-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-top 22043  df-nei 22249
This theorem is referenced by: (None)
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