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Mirrors > Home > MPE Home > Th. List > Mathboxes > gneispa | Structured version Visualization version GIF version |
Description: Each point 𝑝 of the neighborhood space has at least one neighborhood; each neighborhood of 𝑝 contains 𝑝. Axiom A of Seifert And Threlfall. (Contributed by RP, 5-Apr-2021.) |
Ref | Expression |
---|---|
gneispace.x | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
gneispa | ⊢ (𝐽 ∈ Top → ∀𝑝 ∈ 𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4339 | . . . . . . 7 ⊢ (𝑝 ∈ 𝑋 → {𝑝} ⊆ 𝑋) | |
2 | gneispace.x | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | tpnei 20925 | . . . . . . 7 ⊢ (𝐽 ∈ Top → ({𝑝} ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘{𝑝}))) |
4 | 1, 3 | syl5ib 234 | . . . . . 6 ⊢ (𝐽 ∈ Top → (𝑝 ∈ 𝑋 → 𝑋 ∈ ((nei‘𝐽)‘{𝑝}))) |
5 | 4 | imp 445 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘{𝑝})) |
6 | ne0i 3921 | . . . . 5 ⊢ (𝑋 ∈ ((nei‘𝐽)‘{𝑝}) → ((nei‘𝐽)‘{𝑝}) ≠ ∅) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → ((nei‘𝐽)‘{𝑝}) ≠ ∅) |
8 | elnei 20915 | . . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋 ∧ 𝑛 ∈ ((nei‘𝐽)‘{𝑝})) → 𝑝 ∈ 𝑛) | |
9 | 8 | 3expia 1267 | . . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → (𝑛 ∈ ((nei‘𝐽)‘{𝑝}) → 𝑝 ∈ 𝑛)) |
10 | 9 | ralrimiv 2965 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛) |
11 | 7, 10 | jca 554 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑝 ∈ 𝑋) → (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛)) |
12 | 11 | ex 450 | . 2 ⊢ (𝐽 ∈ Top → (𝑝 ∈ 𝑋 → (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛))) |
13 | 12 | ralrimiv 2965 | 1 ⊢ (𝐽 ∈ Top → ∀𝑝 ∈ 𝑋 (((nei‘𝐽)‘{𝑝}) ≠ ∅ ∧ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑝})𝑝 ∈ 𝑛)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∀wral 2912 ⊆ wss 3574 ∅c0 3915 {csn 4177 ∪ cuni 4436 ‘cfv 5888 Topctop 20698 neicnei 20901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-top 20699 df-nei 20902 |
This theorem is referenced by: (None) |
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