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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 21398 | . . . 4 ⊢ (𝐽 ≃ {∅} → 𝐽 ≈ {∅}) | |
2 | df1o2 7459 | . . . 4 ⊢ 1𝑜 = {∅} | |
3 | 1, 2 | syl6breqr 4625 | . . 3 ⊢ (𝐽 ≃ {∅} → 𝐽 ≈ 1𝑜) |
4 | hmphtop1 21392 | . . . 4 ⊢ (𝐽 ≃ {∅} → 𝐽 ∈ Top) | |
5 | en1top 20599 | . . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ≈ 1𝑜 ↔ 𝐽 = {∅})) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐽 ≃ {∅} → (𝐽 ≈ 1𝑜 ↔ 𝐽 = {∅})) |
7 | 3, 6 | mpbid 221 | . 2 ⊢ (𝐽 ≃ {∅} → 𝐽 = {∅}) |
8 | id 22 | . . 3 ⊢ (𝐽 = {∅} → 𝐽 = {∅}) | |
9 | sn0top 20613 | . . . 4 ⊢ {∅} ∈ Top | |
10 | hmphref 21394 | . . . 4 ⊢ ({∅} ∈ Top → {∅} ≃ {∅}) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ {∅} ≃ {∅} |
12 | 8, 11 | syl6eqbr 4622 | . 2 ⊢ (𝐽 = {∅} → 𝐽 ≃ {∅}) |
13 | 7, 12 | impbii 198 | 1 ⊢ (𝐽 ≃ {∅} ↔ 𝐽 = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∅c0 3874 {csn 4125 class class class wbr 4583 1𝑜c1o 7440 ≈ cen 7838 Topctop 20517 ≃ chmph 21367 |
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-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-3or 1032 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-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 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-ord 5643 df-on 5644 df-lim 5645 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-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-1o 7447 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-top 20521 df-topon 20523 df-cn 20841 df-hmeo 21368 df-hmph 21369 |
This theorem is referenced by: hmphindis 21410 |
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