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Theorem gneispb 40359
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 1141 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → (𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})))
21ad2antrr 722 . . . 4 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → (𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})))
3 simpr 485 . . . 4 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → 𝑁𝑠)
4 simplr 765 . . . . 5 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → 𝑠 ∈ 𝒫 𝑋)
54elpwid 4549 . . . 4 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → 𝑠𝑋)
6 gneispace.x . . . . 5 𝑋 = 𝐽
76ssnei2 21652 . . . 4 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ (𝑁𝑠𝑠𝑋)) → 𝑠 ∈ ((nei‘𝐽)‘{𝑃}))
82, 3, 5, 7syl12anc 832 . . 3 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → 𝑠 ∈ ((nei‘𝐽)‘{𝑃}))
98exp31 420 . 2 ((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → (𝑠 ∈ 𝒫 𝑋 → (𝑁𝑠𝑠 ∈ ((nei‘𝐽)‘{𝑃}))))
109ralrimiv 3178 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 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wss 3933  𝒫 cpw 4535  {csn 4557   cuni 4830  cfv 6348  Topctop 21429  neicnei 21633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-top 21430  df-nei 21634
This theorem is referenced by: (None)
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