Theorem isperf3 23069

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 23068	. 2 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
3		dfss3 3919	. . . 4 ⊢ (𝑋 ⊆ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝑋))
4	1	maxlp 23063	. . . . . 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 3154	. . . 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 1541   ∈ wcel 2113  ∀wral 3048   ⊆ wss 3898  {csn 4575  ∪ cuni 4858  ‘cfv 6486  Topctop 22809  limPtclp 23050  Perfcperf 23051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  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 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-top 22810  df-cld 22935  df-ntr 22936  df-cls 22937  df-lp 23052  df-perf 23053

This theorem is referenced by:  perfi  23071  perfopn  23101  t1connperf  23352