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Theorem gneispb 42308
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 ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∀𝑠 ∈ 𝒫 𝑋(𝑁𝑠𝑠 ∈ ((nei‘𝐽)‘{𝑃})))
Distinct variable groups:   𝐽,𝑠   𝑁,𝑠   𝑃,𝑠   𝑋,𝑠

Proof of Theorem gneispb
StepHypRef Expression
1 3simpb 1150 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → (𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})))
21ad2antrr 725 . . . 4 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → (𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})))
3 simpr 486 . . . 4 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → 𝑁𝑠)
4 simplr 768 . . . . 5 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → 𝑠 ∈ 𝒫 𝑋)
54elpwid 4568 . . . 4 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → 𝑠𝑋)
6 gneispace.x . . . . 5 𝑋 = 𝐽
76ssnei2 22419 . . . 4 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ (𝑁𝑠𝑠𝑋)) → 𝑠 ∈ ((nei‘𝐽)‘{𝑃}))
82, 3, 5, 7syl12anc 836 . . 3 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → 𝑠 ∈ ((nei‘𝐽)‘{𝑃}))
98exp31 421 . 2 ((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → (𝑠 ∈ 𝒫 𝑋 → (𝑁𝑠𝑠 ∈ ((nei‘𝐽)‘{𝑃}))))
109ralrimiv 3141 1 ((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → ∀𝑠 ∈ 𝒫 𝑋(𝑁𝑠𝑠 ∈ ((nei‘𝐽)‘{𝑃})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3063  wss 3909  𝒫 cpw 4559  {csn 4585   cuni 4864  cfv 6494  Topctop 22194  neicnei 22400
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-top 22195  df-nei 22401
This theorem is referenced by: (None)
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