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Theorem restsn 23087
Description: The only subspace topology induced by the topology {∅}. (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
Assertion
Ref Expression
restsn (𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})

Proof of Theorem restsn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sn0top 22915 . . . 4 {∅} ∈ Top
2 elrest 17409 . . . 4 (({∅} ∈ Top ∧ 𝐴𝑉) → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴)))
31, 2mpan 689 . . 3 (𝐴𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴)))
4 0ex 5307 . . . . 5 ∅ ∈ V
5 ineq1 4205 . . . . . . 7 (𝑦 = ∅ → (𝑦𝐴) = (∅ ∩ 𝐴))
6 0in 4394 . . . . . . 7 (∅ ∩ 𝐴) = ∅
75, 6eqtrdi 2784 . . . . . 6 (𝑦 = ∅ → (𝑦𝐴) = ∅)
87eqeq2d 2739 . . . . 5 (𝑦 = ∅ → (𝑥 = (𝑦𝐴) ↔ 𝑥 = ∅))
94, 8rexsn 4687 . . . 4 (∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴) ↔ 𝑥 = ∅)
10 velsn 4645 . . . 4 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
119, 10bitr4i 278 . . 3 (∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {∅})
123, 11bitrdi 287 . 2 (𝐴𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ 𝑥 ∈ {∅}))
1312eqrdv 2726 1 (𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1534  wcel 2099  wrex 3067  cin 3946  c0 4323  {csn 4629  (class class class)co 7420  t crest 17402  Topctop 22808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-rest 17404  df-top 22809  df-topon 22826
This theorem is referenced by: (None)
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