Theorem isperf3 23038

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 ∧ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑋

Proof of Theorem isperf3

Step	Hyp	Ref	Expression
1		lpfval.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	isperf2 23037	. 2 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
3		dfss3 3924	. . . 4 ⊢ (𝑋 ⊆ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝑋))
4	1	maxlp 23032	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑥 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑥 ∈ 𝑋 ∧ ¬ {𝑥} ∈ 𝐽)))
5	4	baibd 539	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑋) ↔ ¬ {𝑥} ∈ 𝐽))
6	5	ralbidva 3150	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
7	3, 6	bitrid 283	. . 3 ⊢ (𝐽 ∈ Top → (𝑋 ⊆ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
8	7	pm5.32i 574	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)) ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
9	2, 8	bitri 275	1 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3903  {csn 4577  ∪ cuni 4858  ‘cfv 6482  Topctop 22778  limPtclp 23019  Perfcperf 23020

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-int 4897  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-top 22779  df-cld 22904  df-ntr 22905  df-cls 22906  df-lp 23021  df-perf 23022

This theorem is referenced by:  perfi  23040  perfopn  23070  t1connperf  23321