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Theorem restsn 12131
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 12040 . . . 4 {∅} ∈ Top
2 elrest 11909 . . . 4 (({∅} ∈ Top ∧ 𝐴𝑉) → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴)))
31, 2mpan 418 . . 3 (𝐴𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ ∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴)))
4 0ex 3995 . . . . 5 ∅ ∈ V
5 ineq1 3217 . . . . . . 7 (𝑦 = ∅ → (𝑦𝐴) = (∅ ∩ 𝐴))
6 0in 3345 . . . . . . 7 (∅ ∩ 𝐴) = ∅
75, 6syl6eq 2148 . . . . . 6 (𝑦 = ∅ → (𝑦𝐴) = ∅)
87eqeq2d 2111 . . . . 5 (𝑦 = ∅ → (𝑥 = (𝑦𝐴) ↔ 𝑥 = ∅))
94, 8rexsn 3515 . . . 4 (∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴) ↔ 𝑥 = ∅)
10 velsn 3491 . . . 4 (𝑥 ∈ {∅} ↔ 𝑥 = ∅)
119, 10bitr4i 186 . . 3 (∃𝑦 ∈ {∅}𝑥 = (𝑦𝐴) ↔ 𝑥 ∈ {∅})
123, 11syl6bb 195 . 2 (𝐴𝑉 → (𝑥 ∈ ({∅} ↾t 𝐴) ↔ 𝑥 ∈ {∅}))
1312eqrdv 2098 1 (𝐴𝑉 → ({∅} ↾t 𝐴) = {∅})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104   = wceq 1299  wcel 1448  wrex 2376  cin 3020  c0 3310  {csn 3474  (class class class)co 5706  t crest 11902  Topctop 11946
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-nul 3994  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-rest 11904  df-top 11947  df-topon 11960
This theorem is referenced by: (None)
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