Theorem isperf3 21178

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 21177	. 2 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)))
3		dfss3 3741	. . . 4 ⊢ (𝑋 ⊆ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝑋))
4	1	maxlp 21172	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑥 ∈ ((limPt‘𝐽)‘𝑋) ↔ (𝑥 ∈ 𝑋 ∧ ¬ {𝑥} ∈ 𝐽)))
5	4	baibd 529	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ ((limPt‘𝐽)‘𝑋) ↔ ¬ {𝑥} ∈ 𝐽))
6	5	ralbidva 3134	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝑋 𝑥 ∈ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
7	3, 6	syl5bb 272	. . 3 ⊢ (𝐽 ∈ Top → (𝑋 ⊆ ((limPt‘𝐽)‘𝑋) ↔ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
8	7	pm5.32i 564	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ((limPt‘𝐽)‘𝑋)) ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
9	2, 8	bitri 264	1 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))

Syntax hints:  ¬ wn 3   ↔ wb 196   ∧ wa 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061   ⊆ wss 3723  {csn 4316  ∪ cuni 4574  ‘cfv 6031  Topctop 20918  limPtclp 21159  Perfcperf 21160

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-iin 4657  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-top 20919  df-cld 21044  df-ntr 21045  df-cls 21046  df-lp 21161  df-perf 21162

This theorem is referenced by:  perfi  21180  perfopn  21210  t1connperf  21460