Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  gneispa Structured version   Visualization version   GIF version

Theorem gneispa 44143
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 4808 . . . . . 6 (𝑝𝑋 → {𝑝} ⊆ 𝑋)
2 gneispace.x . . . . . . 7 𝑋 = 𝐽
32tpnei 23129 . . . . . 6 (𝐽 ∈ Top → ({𝑝} ⊆ 𝑋𝑋 ∈ ((nei‘𝐽)‘{𝑝})))
41, 3imbitrid 244 . . . . 5 (𝐽 ∈ Top → (𝑝𝑋𝑋 ∈ ((nei‘𝐽)‘{𝑝})))
54imp 406 . . . 4 ((𝐽 ∈ Top ∧ 𝑝𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝑝}))
65ne0d 4342 . . 3 ((𝐽 ∈ Top ∧ 𝑝𝑋) → ((nei‘𝐽)‘{𝑝}) ≠ ∅)
7 elnei 23119 . . . . 5 ((𝐽 ∈ Top ∧ 𝑝𝑋𝑛 ∈ ((nei‘𝐽)‘{𝑝})) → 𝑝𝑛)
873expia 1122 . . . 4 ((𝐽 ∈ Top ∧ 𝑝𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑝}) → 𝑝𝑛))
98ralrimiv 3145 . . 3 ((𝐽 ∈ Top ∧ 𝑝𝑋) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛)
106, 9jca 511 . 2 ((𝐽 ∈ Top ∧ 𝑝𝑋) → (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛))
1110ralrimiva 3146 1 (𝐽 ∈ Top → ∀𝑝𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝𝑛))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  wral 3061  wss 3951  c0 4333  {csn 4626   cuni 4907  cfv 6561  Topctop 22899  neicnei 23105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-nei 23106
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator