Theorem gneispa 44085

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 → ∀𝑝 ∈ 𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛))

Distinct variable groups:   𝑛,𝐽,𝑝   𝑛,𝑋

Allowed substitution hint:   𝑋(𝑝)

Proof of Theorem gneispa

Step	Hyp	Ref	Expression
1		snssi 4816	. . . . . 6 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ⊆ 𝑋)
2		gneispace.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	tpnei 23127	. . . . . 6 ⊢ (𝐽 ∈ Top → ({𝑝} ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘{𝑝})))
4	1, 3	imbitrid 244	. . . . 5 ⊢ (𝐽 ∈ Top → (𝑝 ∈ 𝑋 → 𝑋 ∈ ((nei‘𝐽)‘{𝑝})))
5	4	imp 406	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝑝}))
6	5	ne0d 4348	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → ((nei‘𝐽)‘{𝑝}) ≠ ∅)
7		elnei 23117	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋 ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝑝})) → 𝑝 ∈ 𝑛)
8	7	3expia 1119	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑝}) → 𝑝 ∈ 𝑛))
9	8	ralrimiv 3141	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛)
10	6, 9	jca 511	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛))
11	10	ralrimiva 3142	1 ⊢ (𝐽 ∈ Top → ∀𝑝 ∈ 𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1535   ∈ wcel 2104   ≠ wne 2936  ∀wral 3057   ⊆ wss 3963  ∅c0 4339  {csn 4631  ∪ cuni 4915  ‘cfv 6559  Topctop 22897  neicnei 23103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3377  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4916  df-iun 5001  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6511  df-fun 6561  df-fn 6562  df-f 6563  df-f1 6564  df-fo 6565  df-f1o 6566  df-fv 6567  df-top 22898  df-nei 23104

This theorem is referenced by: (None)