Theorem gneispa 42871

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 4811	. . . . . 6 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ⊆ 𝑋)
2		gneispace.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	tpnei 22624	. . . . . 6 ⊢ (𝐽 ∈ Top → ({𝑝} ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘{𝑝})))
4	1, 3	imbitrid 243	. . . . 5 ⊢ (𝐽 ∈ Top → (𝑝 ∈ 𝑋 → 𝑋 ∈ ((nei‘𝐽)‘{𝑝})))
5	4	imp 407	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝑝}))
6	5	ne0d 4335	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → ((nei‘𝐽)‘{𝑝}) ≠ ∅)
7		elnei 22614	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋 ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝑝})) → 𝑝 ∈ 𝑛)
8	7	3expia 1121	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑝}) → 𝑝 ∈ 𝑛))
9	8	ralrimiv 3145	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛)
10	6, 9	jca 512	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛))
11	10	ralrimiva 3146	1 ⊢ (𝐽 ∈ Top → ∀𝑝 ∈ 𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061   ⊆ wss 3948  ∅c0 4322  {csn 4628  ∪ cuni 4908  ‘cfv 6543  Topctop 22394  neicnei 22600

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-top 22395  df-nei 22601

This theorem is referenced by: (None)