Theorem gneispb 41741

Description: Given a neighborhood 𝑁 of 𝑃, each subset of the neighborhood space containing this neighborhood is also a neighborhood of 𝑃. Axiom B of Seifert and Threlfall. (Contributed by RP, 5-Apr-2021.)

Hypothesis

Ref	Expression
gneispace.x	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
gneispb	⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∀𝑠 ∈ 𝒫 𝑋(𝑁 ⊆ 𝑠 → 𝑠 ∈ ((nei‘𝐽)‘{𝑃})))

Distinct variable groups:   𝐽,𝑠   𝑁,𝑠   𝑃,𝑠   𝑋,𝑠

Proof of Theorem gneispb

Step	Hyp	Ref	Expression
1		3simpb 1148	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → (𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})))
2	1	ad2antrr 723	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁 ⊆ 𝑠) → (𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})))
3		simpr 485	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁 ⊆ 𝑠) → 𝑁 ⊆ 𝑠)
4		simplr 766	. . . . 5 ⊢ ((((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁 ⊆ 𝑠) → 𝑠 ∈ 𝒫 𝑋)
5	4	elpwid 4544	. . . 4 ⊢ ((((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁 ⊆ 𝑠) → 𝑠 ⊆ 𝑋)
6		gneispace.x	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
7	6	ssnei2 22267	. . . 4 ⊢ (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ (𝑁 ⊆ 𝑠 ∧ 𝑠 ⊆ 𝑋)) → 𝑠 ∈ ((nei‘𝐽)‘{𝑃}))
8	2, 3, 5, 7	syl12anc 834	. . 3 ⊢ ((((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁 ⊆ 𝑠) → 𝑠 ∈ ((nei‘𝐽)‘{𝑃}))
9	8	exp31 420	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → (𝑠 ∈ 𝒫 𝑋 → (𝑁 ⊆ 𝑠 → 𝑠 ∈ ((nei‘𝐽)‘{𝑃}))))
10	9	ralrimiv 3102	1 ⊢ ((𝐽 ∈ Top ∧ 𝑃 ∈ 𝑋 ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∀𝑠 ∈ 𝒫 𝑋(𝑁 ⊆ 𝑠 → 𝑠 ∈ ((nei‘𝐽)‘{𝑃})))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887  𝒫 cpw 4533  {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)