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Theorem t1conperf 21049
 Description: A connected T1 space is perfect, unless it is the topology of a singleton. (Contributed by Mario Carneiro, 26-Dec-2016.)
Hypothesis
Ref Expression
t1conperf.1 𝑋 = 𝐽
Assertion
Ref Expression
t1conperf ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con ∧ ¬ 𝑋 ≈ 1𝑜) → 𝐽 ∈ Perf)

Proof of Theorem t1conperf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 t1conperf.1 . . . . . . . 8 𝑋 = 𝐽
2 simplr 788 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝐽 ∈ Con)
3 simprr 792 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ 𝐽)
4 vex 3176 . . . . . . . . . 10 𝑥 ∈ V
54snnz 4252 . . . . . . . . 9 {𝑥} ≠ ∅
65a1i 11 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ≠ ∅)
71t1sncld 20940 . . . . . . . . 9 ((𝐽 ∈ Fre ∧ 𝑥𝑋) → {𝑥} ∈ (Clsd‘𝐽))
87ad2ant2r 779 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ (Clsd‘𝐽))
91, 2, 3, 6, 8conclo 21028 . . . . . . 7 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} = 𝑋)
104ensn1 7906 . . . . . . 7 {𝑥} ≈ 1𝑜
119, 10syl6eqbrr 4623 . . . . . 6 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝑋 ≈ 1𝑜)
1211rexlimdvaa 3014 . . . . 5 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) → (∃𝑥𝑋 {𝑥} ∈ 𝐽𝑋 ≈ 1𝑜))
1312con3d 147 . . . 4 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) → (¬ 𝑋 ≈ 1𝑜 → ¬ ∃𝑥𝑋 {𝑥} ∈ 𝐽))
14 ralnex 2975 . . . 4 (∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽 ↔ ¬ ∃𝑥𝑋 {𝑥} ∈ 𝐽)
1513, 14syl6ibr 241 . . 3 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) → (¬ 𝑋 ≈ 1𝑜 → ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
16 t1top 20944 . . . . 5 (𝐽 ∈ Fre → 𝐽 ∈ Top)
1716adantr 480 . . . 4 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) → 𝐽 ∈ Top)
181isperf3 20767 . . . . 5 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
1918baib 942 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Perf ↔ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
2017, 19syl 17 . . 3 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) → (𝐽 ∈ Perf ↔ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
2115, 20sylibrd 248 . 2 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con) → (¬ 𝑋 ≈ 1𝑜𝐽 ∈ Perf))
22213impia 1253 1 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Con ∧ ¬ 𝑋 ≈ 1𝑜) → 𝐽 ∈ Perf)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  ∅c0 3874  {csn 4125  ∪ cuni 4372   class class class wbr 4583  ‘cfv 5804  1𝑜c1o 7440   ≈ cen 7838  Topctop 20517  Clsdccld 20630  Perfcperf 20749  Frect1 20921  Conccon 21024 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-1o 7447  df-en 7842  df-top 20521  df-cld 20633  df-ntr 20634  df-cls 20635  df-lp 20750  df-perf 20751  df-t1 20928  df-con 21025 This theorem is referenced by: (None)
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