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Theorem gneispace0nelrn3 40498
 Description: A generic neighborhood space has a nonempty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.)
Hypothesis
Ref Expression
gneispace.a 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}
Assertion
Ref Expression
gneispace0nelrn3 (𝐹𝐴 → ¬ ∅ ∈ ran 𝐹)
Distinct variable groups:   𝑛,𝐹,𝑝,𝑓   𝐹,𝑠,𝑓   𝑓,𝑛,𝑝
Allowed substitution hints:   𝐴(𝑓,𝑛,𝑠,𝑝)

Proof of Theorem gneispace0nelrn3
StepHypRef Expression
1 gneispace.a . . 3 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓𝑛 ∈ (𝑓𝑝)(𝑝𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛𝑠𝑠 ∈ (𝑓𝑝))))}
21gneispacern 40494 . 2 (𝐹𝐴 → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
3 neldifsnd 4729 . . 3 (ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) → ¬ ∅ ∈ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
4 ssel 3964 . . 3 (ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) → (∅ ∈ ran 𝐹 → ∅ ∈ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})))
53, 4mtod 200 . 2 (ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) → ¬ ∅ ∈ ran 𝐹)
62, 5syl 17 1 (𝐹𝐴 → ¬ ∅ ∈ ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113  {cab 2802  ∀wral 3141   ∖ cdif 3936   ⊆ wss 3939  ∅c0 4294  𝒫 cpw 4542  {csn 4570  dom cdm 5558  ran crn 5559  ⟶wf 6354  ‘cfv 6358 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rab 3150  df-v 3499  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366 This theorem is referenced by: (None)
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