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Theorem isperf3 21761
Description: A perfect space is a topology which has no open singletons. (Contributed by Mario Carneiro, 24-Dec-2016.)
Hypothesis
Ref Expression
lpfval.1 𝑋 = 𝐽
Assertion
Ref Expression
isperf3 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem isperf3
StepHypRef Expression
1 lpfval.1 . . 3 𝑋 = 𝐽
21isperf2 21760 . 2 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
3 dfss3 3941 . . . 4 (𝑋 ⊆ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝑋))
41maxlp 21755 . . . . . 6 (𝐽 ∈ Top → (𝑥 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑥𝑋 ∧ ¬ {𝑥} ∈ 𝐽)))
54baibd 543 . . . . 5 ((𝐽 ∈ Top ∧ 𝑥𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑋) ↔ ¬ {𝑥} ∈ 𝐽))
65ralbidva 3191 . . . 4 (𝐽 ∈ Top → (∀𝑥𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
73, 6syl5bb 286 . . 3 (𝐽 ∈ Top → (𝑋 ⊆ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
87pm5.32i 578 . 2 ((𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)) ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
92, 8bitri 278 1 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  wss 3919  {csn 4550   cuni 4824  cfv 6343  Topctop 21501  limPtclp 21742  Perfcperf 21743
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-top 21502  df-cld 21627  df-ntr 21628  df-cls 21629  df-lp 21744  df-perf 21745
This theorem is referenced by:  perfi  21763  perfopn  21793  t1connperf  22044
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