Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  gneispb Structured version   Visualization version   GIF version

Theorem gneispb 37449
 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 1052 . . . . 5 ((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → (𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})))
21ad2antrr 758 . . . 4 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → (𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})))
3 simpr 476 . . . 4 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → 𝑁𝑠)
4 simplr 788 . . . . 5 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → 𝑠 ∈ 𝒫 𝑋)
54elpwid 4118 . . . 4 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → 𝑠𝑋)
6 gneispace.x . . . . 5 𝑋 = 𝐽
76ssnei2 20730 . . . 4 (((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ (𝑁𝑠𝑠𝑋)) → 𝑠 ∈ ((nei‘𝐽)‘{𝑃}))
82, 3, 5, 7syl12anc 1316 . . 3 ((((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) ∧ 𝑠 ∈ 𝒫 𝑋) ∧ 𝑁𝑠) → 𝑠 ∈ ((nei‘𝐽)‘{𝑃}))
98exp31 628 . 2 ((𝐽 ∈ Top ∧ 𝑃𝑋𝑁 ∈ ((nei‘𝐽)‘{𝑃})) → (𝑠 ∈ 𝒫 𝑋 → (𝑁𝑠𝑠 ∈ ((nei‘𝐽)‘{𝑃}))))
109ralrimiv 2948 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 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  neicnei 20711 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-top 20521  df-nei 20712 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator