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| Mirrors > Home > MPE Home > Th. List > t1connperf | 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 | 
|---|---|
| t1connperf.1 | ⊢ 𝑋 = ∪ 𝐽 | 
| Ref | Expression | 
|---|---|
| t1connperf | ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn ∧ ¬ 𝑋 ≈ 1o) → 𝐽 ∈ Perf) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | t1connperf.1 | . . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽 | |
| 2 | simplr 768 | . . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝐽 ∈ Conn) | |
| 3 | simprr 772 | . . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ 𝐽) | |
| 4 | vex 3483 | . . . . . . . . . 10 ⊢ 𝑥 ∈ V | |
| 5 | 4 | snnz 4775 | . . . . . . . . 9 ⊢ {𝑥} ≠ ∅ | 
| 6 | 5 | a1i 11 | . . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ≠ ∅) | 
| 7 | 1 | t1sncld 23335 | . . . . . . . . 9 ⊢ ((𝐽 ∈ Fre ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ (Clsd‘𝐽)) | 
| 8 | 7 | ad2ant2r 747 | . . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ (Clsd‘𝐽)) | 
| 9 | 1, 2, 3, 6, 8 | connclo 23424 | . . . . . . 7 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} = 𝑋) | 
| 10 | 4 | ensn1 9062 | . . . . . . 7 ⊢ {𝑥} ≈ 1o | 
| 11 | 9, 10 | eqbrtrrdi 5182 | . . . . . 6 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝑋 ≈ 1o) | 
| 12 | 11 | rexlimdvaa 3155 | . . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (∃𝑥 ∈ 𝑋 {𝑥} ∈ 𝐽 → 𝑋 ≈ 1o)) | 
| 13 | 12 | con3d 152 | . . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ¬ ∃𝑥 ∈ 𝑋 {𝑥} ∈ 𝐽)) | 
| 14 | ralnex 3071 | . . . 4 ⊢ (∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽 ↔ ¬ ∃𝑥 ∈ 𝑋 {𝑥} ∈ 𝐽) | |
| 15 | 13, 14 | imbitrrdi 252 | . . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽)) | 
| 16 | t1top 23339 | . . . . 5 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top) | |
| 17 | 16 | adantr 480 | . . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → 𝐽 ∈ Top) | 
| 18 | 1 | isperf3 23162 | . . . . 5 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽)) | 
| 19 | 18 | baib 535 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Perf ↔ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽)) | 
| 20 | 17, 19 | syl 17 | . . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (𝐽 ∈ Perf ↔ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽)) | 
| 21 | 15, 20 | sylibrd 259 | . 2 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → 𝐽 ∈ Perf)) | 
| 22 | 21 | 3impia 1117 | 1 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn ∧ ¬ 𝑋 ≈ 1o) → 𝐽 ∈ Perf) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 ∅c0 4332 {csn 4625 ∪ cuni 4906 class class class wbr 5142 ‘cfv 6560 1oc1o 8500 ≈ cen 8983 Topctop 22900 Clsdccld 23025 Perfcperf 23144 Frect1 23316 Conncconn 23420 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-1o 8507 df-en 8987 df-top 22901 df-cld 23028 df-ntr 23029 df-cls 23030 df-lp 23145 df-perf 23146 df-t1 23323 df-conn 23421 | 
| This theorem is referenced by: (None) | 
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