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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 23136 | . . . 4 ⊢ (𝐽 ≃ {∅} → 𝐽 ≈ {∅}) | |
2 | df1o2 8419 | . . . 4 ⊢ 1o = {∅} | |
3 | 1, 2 | breqtrrdi 5147 | . . 3 ⊢ (𝐽 ≃ {∅} → 𝐽 ≈ 1o) |
4 | hmphtop1 23130 | . . . 4 ⊢ (𝐽 ≃ {∅} → 𝐽 ∈ Top) | |
5 | en1top 22334 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ≈ 1o ↔ 𝐽 = {∅})) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐽 ≃ {∅} → (𝐽 ≈ 1o ↔ 𝐽 = {∅})) |
7 | 3, 6 | mpbid 231 | . 2 ⊢ (𝐽 ≃ {∅} → 𝐽 = {∅}) |
8 | id 22 | . . 3 ⊢ (𝐽 = {∅} → 𝐽 = {∅}) | |
9 | sn0top 22349 | . . . 4 ⊢ {∅} ∈ Top | |
10 | hmphref 23132 | . . . 4 ⊢ ({∅} ∈ Top → {∅} ≃ {∅}) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ {∅} ≃ {∅} |
12 | 8, 11 | eqbrtrdi 5144 | . 2 ⊢ (𝐽 = {∅} → 𝐽 ≃ {∅}) |
13 | 7, 12 | impbii 208 | 1 ⊢ (𝐽 ≃ {∅} ↔ 𝐽 = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∈ wcel 2106 ∅c0 4282 {csn 4586 class class class wbr 5105 1oc1o 8405 ≈ cen 8880 Topctop 22242 ≃ chmph 23105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-1o 8412 df-map 8767 df-en 8884 df-top 22243 df-topon 22260 df-cn 22578 df-hmeo 23106 df-hmph 23107 |
This theorem is referenced by: hmphindis 23148 |
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