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Mirrors > Home > MPE Home > Th. List > t1ficld | Structured version Visualization version GIF version |
Description: In a T1 space, finite sets are closed. (Contributed by Mario Carneiro, 25-Dec-2016.) |
Ref | Expression |
---|---|
ist0.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
t1ficld | ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ⊆ 𝑋 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunid 5064 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑥} = 𝐴 | |
2 | ist0.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 2 | ist1 22825 | . . . . 5 ⊢ (𝐽 ∈ Fre ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽))) |
4 | 3 | simplbi 499 | . . . 4 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top) |
5 | 4 | 3ad2ant1 1134 | . . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ⊆ 𝑋 ∧ 𝐴 ∈ Fin) → 𝐽 ∈ Top) |
6 | simp3 1139 | . . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ⊆ 𝑋 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ Fin) | |
7 | 3 | simprbi 498 | . . . . 5 ⊢ (𝐽 ∈ Fre → ∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽)) |
8 | ssralv 4051 | . . . . 5 ⊢ (𝐴 ⊆ 𝑋 → (∀𝑥 ∈ 𝑋 {𝑥} ∈ (Clsd‘𝐽) → ∀𝑥 ∈ 𝐴 {𝑥} ∈ (Clsd‘𝐽))) | |
9 | 7, 8 | mpan9 508 | . . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ⊆ 𝑋) → ∀𝑥 ∈ 𝐴 {𝑥} ∈ (Clsd‘𝐽)) |
10 | 9 | 3adant3 1133 | . . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ⊆ 𝑋 ∧ 𝐴 ∈ Fin) → ∀𝑥 ∈ 𝐴 {𝑥} ∈ (Clsd‘𝐽)) |
11 | 2 | iuncld 22549 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ Fin ∧ ∀𝑥 ∈ 𝐴 {𝑥} ∈ (Clsd‘𝐽)) → ∪ 𝑥 ∈ 𝐴 {𝑥} ∈ (Clsd‘𝐽)) |
12 | 5, 6, 10, 11 | syl3anc 1372 | . 2 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ⊆ 𝑋 ∧ 𝐴 ∈ Fin) → ∪ 𝑥 ∈ 𝐴 {𝑥} ∈ (Clsd‘𝐽)) |
13 | 1, 12 | eqeltrrid 2839 | 1 ⊢ ((𝐽 ∈ Fre ∧ 𝐴 ⊆ 𝑋 ∧ 𝐴 ∈ Fin) → 𝐴 ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ⊆ wss 3949 {csn 4629 ∪ cuni 4909 ∪ ciun 4998 ‘cfv 6544 Fincfn 8939 Topctop 22395 Clsdccld 22520 Frect1 22811 |
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-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-om 7856 df-1st 7975 df-2nd 7976 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-fin 8943 df-top 22396 df-cld 22523 df-t1 22818 |
This theorem is referenced by: poimirlem30 36566 |
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