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Theorem hmph0 23804
Description: A topology homeomorphic to the empty set is empty. (Contributed by FL, 18-Aug-2008.) (Revised by Mario Carneiro, 10-Sep-2015.)
Assertion
Ref Expression
hmph0 (𝐽 ≃ {∅} ↔ 𝐽 = {∅})

Proof of Theorem hmph0
StepHypRef Expression
1 hmphen 23794 . . . 4 (𝐽 ≃ {∅} → 𝐽 ≈ {∅})
2 df1o2 8514 . . . 4 1o = {∅}
31, 2breqtrrdi 5184 . . 3 (𝐽 ≃ {∅} → 𝐽 ≈ 1o)
4 hmphtop1 23788 . . . 4 (𝐽 ≃ {∅} → 𝐽 ∈ Top)
5 en1top 22992 . . . 4 (𝐽 ∈ Top → (𝐽 ≈ 1o𝐽 = {∅}))
64, 5syl 17 . . 3 (𝐽 ≃ {∅} → (𝐽 ≈ 1o𝐽 = {∅}))
73, 6mpbid 232 . 2 (𝐽 ≃ {∅} → 𝐽 = {∅})
8 id 22 . . 3 (𝐽 = {∅} → 𝐽 = {∅})
9 sn0top 23007 . . . 4 {∅} ∈ Top
10 hmphref 23790 . . . 4 ({∅} ∈ Top → {∅} ≃ {∅})
119, 10ax-mp 5 . . 3 {∅} ≃ {∅}
128, 11eqbrtrdi 5181 . 2 (𝐽 = {∅} → 𝐽 ≃ {∅})
137, 12impbii 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
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