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Theorem hmph0 23146
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 23136 . . . 4 (𝐽 ≃ {∅} → 𝐽 ≈ {∅})
2 df1o2 8419 . . . 4 1o = {∅}
31, 2breqtrrdi 5147 . . 3 (𝐽 ≃ {∅} → 𝐽 ≈ 1o)
4 hmphtop1 23130 . . . 4 (𝐽 ≃ {∅} → 𝐽 ∈ Top)
5 en1top 22334 . . . 4 (𝐽 ∈ Top → (𝐽 ≈ 1o𝐽 = {∅}))
64, 5syl 17 . . 3 (𝐽 ≃ {∅} → (𝐽 ≈ 1o𝐽 = {∅}))
73, 6mpbid 231 . 2 (𝐽 ≃ {∅} → 𝐽 = {∅})
8 id 22 . . 3 (𝐽 = {∅} → 𝐽 = {∅})
9 sn0top 22349 . . . 4 {∅} ∈ Top
10 hmphref 23132 . . . 4 ({∅} ∈ Top → {∅} ≃ {∅})
119, 10ax-mp 5 . . 3 {∅} ≃ {∅}
128, 11eqbrtrdi 5144 . 2 (𝐽 = {∅} → 𝐽 ≃ {∅})
137, 12impbii 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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