Theorem gneispa 44367

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 4764	. . . . . 6 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ⊆ 𝑋)
2		gneispace.x	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	tpnei 23065	. . . . . 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 4294	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → ((nei‘𝐽)‘{𝑝}) ≠ ∅)
7		elnei 23055	. . . . 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 3127	. . 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 3128	1 ⊢ (𝐽 ∈ Top → ∀𝑝 ∈ 𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051   ⊆ wss 3901  ∅c0 4285  {csn 4580  ∪ cuni 4863  ‘cfv 6492  Topctop 22837  neicnei 23041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-top 22838  df-nei 23042

This theorem is referenced by: (None)