Theorem gneispace0nelrn3 44132

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 44128	. 2 ⊢ (𝐹 ∈ 𝐴 → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
3		neldifsnd 4773	. . 3 ⊢ (ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) → ¬ ∅ ∈ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}))
4		ssel 3957	. . 3 ⊢ (ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) → (∅ ∈ ran 𝐹 → ∅ ∈ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})))
5	3, 4	mtod 198	. 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 395   = wceq 1539   ∈ wcel 2107  {cab 2712  ∀wral 3050   ∖ cdif 3928   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  {csn 4606  dom cdm 5665  ran crn 5666  ⟶wf 6537  ‘cfv 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549

This theorem is referenced by: (None)