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Mirrors > Home > MPE Home > Th. List > islp2 | Structured version Visualization version GIF version |
Description: The predicate "𝑃 is a limit point of 𝑆 " in terms of neighborhoods. Definition of limit point in [Munkres] p. 97. Although Munkres uses open neighborhoods, it also works for our more general neighborhoods. (Contributed by NM, 26-Feb-2007.) (Proof shortened by Mario Carneiro, 25-Dec-2016.) |
Ref | Expression |
---|---|
lpfval.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
islp2 | ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lpfval.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | islp 21731 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})))) |
3 | 2 | 3adant3 1128 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ 𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})))) |
4 | ssdifss 4100 | . . 3 ⊢ (𝑆 ⊆ 𝑋 → (𝑆 ∖ {𝑃}) ⊆ 𝑋) | |
5 | 1 | neindisj2 21714 | . . 3 ⊢ ((𝐽 ∈ Top ∧ (𝑆 ∖ {𝑃}) ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)) |
6 | 4, 5 | syl3an2 1160 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((cls‘𝐽)‘(𝑆 ∖ {𝑃})) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)) |
7 | 3, 6 | bitrd 281 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ 𝑋 ∧ 𝑃 ∈ 𝑋) → (𝑃 ∈ ((limPt‘𝐽)‘𝑆) ↔ ∀𝑛 ∈ ((nei‘𝐽)‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∖ cdif 3921 ∩ cin 3923 ⊆ wss 3924 ∅c0 4279 {csn 4553 ∪ cuni 4824 ‘cfv 6341 Topctop 21484 clsccl 21609 neicnei 21688 limPtclp 21725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-iin 4908 df-br 5053 df-opab 5115 df-mpt 5133 df-id 5446 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-top 21485 df-cld 21610 df-ntr 21611 df-cls 21612 df-nei 21689 df-lp 21727 |
This theorem is referenced by: clslp 21739 lpbl 23096 reperflem 23409 islptre 41990 islpcn 42010 |
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