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Mirrors > Home > MPE Home > Th. List > Mathboxes > gneispace0nelrn3 | Structured version Visualization version GIF version |
Description: A generic neighborhood space has a nonempty set of neighborhoods for every point in its domain. (Contributed by RP, 15-Apr-2021.) |
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))))} |
Ref | Expression |
---|---|
gneispace0nelrn3 | ⊢ (𝐹 ∈ 𝐴 → ¬ ∅ ∈ ran 𝐹) |
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 44086 | . 2 ⊢ (𝐹 ∈ 𝐴 → ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) |
3 | neldifsnd 4800 | . . 3 ⊢ (ran 𝐹 ⊆ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅}) → ¬ ∅ ∈ (𝒫 (𝒫 dom 𝐹 ∖ {∅}) ∖ {∅})) | |
4 | ssel 3989 | . . 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) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1535 ∈ wcel 2104 {cab 2710 ∀wral 3057 ∖ cdif 3960 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4604 {csn 4630 dom cdm 5683 ran crn 5684 ⟶wf 6554 ‘cfv 6558 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-ne 2937 df-ral 3058 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-fv 6566 |
This theorem is referenced by: (None) |
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