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Mirrors > Home > MPE Home > Th. List > t1conperf | Structured version Visualization version GIF version |
Description: A connected T1 space is perfect, unless it is the topology of a singleton. (Contributed by Mario Carneiro, 26-Dec-2016.) |
Ref | Expression |
---|---|
t1conperf.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
t1conperf | ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con ∧ ¬ 𝑋 ≈ 1𝑜) → 𝐽 ∈ Perf) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | t1conperf.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
2 | simplr 788 | . . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝐽 ∈ Con) | |
3 | simprr 792 | . . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ 𝐽) | |
4 | vex 3176 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
5 | 4 | snnz 4252 | . . . . . . . . 9 ⊢ {𝑥} ≠ ∅ |
6 | 5 | a1i 11 | . . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ≠ ∅) |
7 | 1 | t1sncld 20940 | . . . . . . . . 9 ⊢ ((𝐽 ∈ Fre ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ (Clsd‘𝐽)) |
8 | 7 | ad2ant2r 779 | . . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ (Clsd‘𝐽)) |
9 | 1, 2, 3, 6, 8 | conclo 21028 | . . . . . . 7 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} = 𝑋) |
10 | 4 | ensn1 7906 | . . . . . . 7 ⊢ {𝑥} ≈ 1𝑜 |
11 | 9, 10 | syl6eqbrr 4623 | . . . . . 6 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝑋 ≈ 1𝑜) |
12 | 11 | rexlimdvaa 3014 | . . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) → (∃𝑥 ∈ 𝑋 {𝑥} ∈ 𝐽 → 𝑋 ≈ 1𝑜)) |
13 | 12 | con3d 147 | . . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) → (¬ 𝑋 ≈ 1𝑜 → ¬ ∃𝑥 ∈ 𝑋 {𝑥} ∈ 𝐽)) |
14 | ralnex 2975 | . . . 4 ⊢ (∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽 ↔ ¬ ∃𝑥 ∈ 𝑋 {𝑥} ∈ 𝐽) | |
15 | 13, 14 | syl6ibr 241 | . . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) → (¬ 𝑋 ≈ 1𝑜 → ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽)) |
16 | t1top 20944 | . . . . 5 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top) | |
17 | 16 | adantr 480 | . . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) → 𝐽 ∈ Top) |
18 | 1 | isperf3 20767 | . . . . 5 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽)) |
19 | 18 | baib 942 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Perf ↔ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽)) |
20 | 17, 19 | syl 17 | . . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) → (𝐽 ∈ Perf ↔ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽)) |
21 | 15, 20 | sylibrd 248 | . 2 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) → (¬ 𝑋 ≈ 1𝑜 → 𝐽 ∈ Perf)) |
22 | 21 | 3impia 1253 | 1 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con ∧ ¬ 𝑋 ≈ 1𝑜) → 𝐽 ∈ Perf) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 ∅c0 3874 {csn 4125 ∪ cuni 4372 class class class wbr 4583 ‘cfv 5804 1𝑜c1o 7440 ≈ cen 7838 Topctop 20517 Clsdccld 20630 Perfcperf 20749 Frect1 20921 Conccon 21024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-1o 7447 df-en 7842 df-top 20521 df-cld 20633 df-ntr 20634 df-cls 20635 df-lp 20750 df-perf 20751 df-t1 20928 df-con 21025 |
This theorem is referenced by: (None) |
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