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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 23809 | . . . 4 ⊢ (𝐽 ≃ {∅} → 𝐽 ≈ {∅}) | |
2 | df1o2 8512 | . . . 4 ⊢ 1o = {∅} | |
3 | 1, 2 | breqtrrdi 5190 | . . 3 ⊢ (𝐽 ≃ {∅} → 𝐽 ≈ 1o) |
4 | hmphtop1 23803 | . . . 4 ⊢ (𝐽 ≃ {∅} → 𝐽 ∈ Top) | |
5 | en1top 23007 | . . . 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 23022 | . . . 4 ⊢ {∅} ∈ Top | |
10 | hmphref 23805 | . . . 4 ⊢ ({∅} ∈ Top → {∅} ≃ {∅}) | |
11 | 9, 10 | ax-mp 5 | . . 3 ⊢ {∅} ≃ {∅} |
12 | 8, 11 | eqbrtrdi 5187 | . 2 ⊢ (𝐽 = {∅} → 𝐽 ≃ {∅}) |
13 | 7, 12 | impbii 209 | 1 ⊢ (𝐽 ≃ {∅} ↔ 𝐽 = {∅}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∈ wcel 2106 ∅c0 4339 {csn 4631 class class class wbr 5148 1oc1o 8498 ≈ cen 8981 Topctop 22915 ≃ chmph 23778 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-1o 8505 df-map 8867 df-en 8985 df-top 22916 df-topon 22933 df-cn 23251 df-hmeo 23779 df-hmph 23780 |
This theorem is referenced by: hmphindis 23821 |
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