Theorem gneispace0nelrn3 42893

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 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}

Assertion

Ref	Expression
gneispace0nelrn3	⊢ (𝐹 ∈ 𝐴 → ¬ ∅ ∈ 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

Step	Hyp	Ref	Expression
1		gneispace.a	. . 3 ⊢ 𝐴 = {𝑓 ∣ (𝑓:dom 𝑓⟶(𝒫 (𝒫 dom 𝑓 ∖ {∅}) ∖ {∅}) ∧ ∀𝑝 ∈ dom 𝑓∀𝑛 ∈ (𝑓‘𝑝)(𝑝 ∈ 𝑛 ∧ ∀𝑠 ∈ 𝒫 dom 𝑓(𝑛 ⊆ 𝑠 → 𝑠 ∈ (𝑓‘𝑝))))}
2	1	gneispacern 42889	. 2 ⊢ (𝐹 ∈ 𝐴 → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
3		neldifsnd 4797	. . 3 ⊢ (ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) → ¬ ∅ ∈ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
4		ssel 3976	. . 3 ⊢ (ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) → (∅ ∈ ran 𝐹 → ∅ ∈ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})))
5	3, 4	mtod 197	. 2 ⊢ (ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) → ¬ ∅ ∈ ran 𝐹)
6	2, 5	syl 17	1 ⊢ (𝐹 ∈ 𝐴 → ¬ ∅ ∈ ran 𝐹)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  ∀wral 3062   ∖ cdif 3946   ⊆ wss 3949  ∅c0 4323  𝒫 cpw 4603  {csn 4629  dom cdm 5677  ran crn 5678  ⟶wf 6540  ‘cfv 6544

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-ext 2704

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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552

This theorem is referenced by: (None)