|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > hmph0 | Structured version Visualization version GIF version | ||
| Description: A topology homeomorphic to the empty set is empty. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) | 
| Ref | Expression | 
|---|---|
| hmph0 | ⊢ (𝐽 ≃ {∅} ↔ 𝐽 = {∅}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | hmphen 23794 | . . . 4 ⊢ (𝐽 ≃ {∅} → 𝐽 ≈ {∅}) | |
| 2 | df1o2 8514 | . . . 4 ⊢ 1o = {∅} | |
| 3 | 1, 2 | breqtrrdi 5184 | . . 3 ⊢ (𝐽 ≃ {∅} → 𝐽 ≈ 1o) | 
| 4 | hmphtop1 23788 | . . . 4 ⊢ (𝐽 ≃ {∅} → 𝐽 ∈ Top) | |
| 5 | en1top 22992 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ≈ 1o ↔ 𝐽 = {∅})) | |
| 6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐽 ≃ {∅} → (𝐽 ≈ 1o ↔ 𝐽 = {∅})) | 
| 7 | 3, 6 | mpbid 232 | . 2 ⊢ (𝐽 ≃ {∅} → 𝐽 = {∅}) | 
| 8 | id 22 | . . 3 ⊢ (𝐽 = {∅} → 𝐽 = {∅}) | |
| 9 | sn0top 23007 | . . . 4 ⊢ {∅} ∈ Top | |
| 10 | hmphref 23790 | . . . 4 ⊢ ({∅} ∈ Top → {∅} ≃ {∅}) | |
| 11 | 9, 10 | ax-mp 5 | . . 3 ⊢ {∅} ≃ {∅} | 
| 12 | 8, 11 | eqbrtrdi 5181 | . 2 ⊢ (𝐽 = {∅} → 𝐽 ≃ {∅}) | 
| 13 | 7, 12 | impbii 209 | 1 ⊢ (𝐽 ≃ {∅} ↔ 𝐽 = {∅}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 ∈ wcel 2107 ∅c0 4332 {csn 4625 class class class wbr 5142 1oc1o 8500 ≈ cen 8983 Topctop 22900 ≃ chmph 23763 | 
| 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-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-iun 4992 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-ov 7435 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 df-1o 8507 df-map 8869 df-en 8987 df-top 22901 df-topon 22918 df-cn 23236 df-hmeo 23764 df-hmph 23765 | 
| This theorem is referenced by: hmphindis 23806 | 
| Copyright terms: Public domain | W3C validator |