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Theorem hmph0 23819
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 23809 . . . 4 (𝐽 ≃ {∅} → 𝐽 ≈ {∅})
2 df1o2 8512 . . . 4 1o = {∅}
31, 2breqtrrdi 5190 . . 3 (𝐽 ≃ {∅} → 𝐽 ≈ 1o)
4 hmphtop1 23803 . . . 4 (𝐽 ≃ {∅} → 𝐽 ∈ Top)
5 en1top 23007 . . . 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 23022 . . . 4 {∅} ∈ Top
10 hmphref 23805 . . . 4 ({∅} ∈ Top → {∅} ≃ {∅})
119, 10ax-mp 5 . . 3 {∅} ≃ {∅}
128, 11eqbrtrdi 5187 . 2 (𝐽 = {∅} → 𝐽 ≃ {∅})
137, 12impbii 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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