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Theorem t1connperf 23358
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
t1connperf.1 𝑋 = 𝐽
Assertion
Ref Expression
t1connperf ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn ∧ ¬ 𝑋 ≈ 1o) → 𝐽 ∈ Perf)

Proof of Theorem t1connperf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 t1connperf.1 . . . . . . . 8 𝑋 = 𝐽
2 simplr 767 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝐽 ∈ Conn)
3 simprr 771 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ 𝐽)
4 vex 3467 . . . . . . . . . 10 𝑥 ∈ V
54snnz 4776 . . . . . . . . 9 {𝑥} ≠ ∅
65a1i 11 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ≠ ∅)
71t1sncld 23248 . . . . . . . . 9 ((𝐽 ∈ Fre ∧ 𝑥𝑋) → {𝑥} ∈ (Clsd‘𝐽))
87ad2ant2r 745 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ (Clsd‘𝐽))
91, 2, 3, 6, 8connclo 23337 . . . . . . 7 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} = 𝑋)
104ensn1 9040 . . . . . . 7 {𝑥} ≈ 1o
119, 10eqbrtrrdi 5183 . . . . . 6 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝑋 ≈ 1o)
1211rexlimdvaa 3146 . . . . 5 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (∃𝑥𝑋 {𝑥} ∈ 𝐽𝑋 ≈ 1o))
1312con3d 152 . . . 4 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ¬ ∃𝑥𝑋 {𝑥} ∈ 𝐽))
14 ralnex 3062 . . . 4 (∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽 ↔ ¬ ∃𝑥𝑋 {𝑥} ∈ 𝐽)
1513, 14imbitrrdi 251 . . 3 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
16 t1top 23252 . . . . 5 (𝐽 ∈ Fre → 𝐽 ∈ Top)
1716adantr 479 . . . 4 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → 𝐽 ∈ Top)
181isperf3 23075 . . . . 5 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
1918baib 534 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Perf ↔ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
2017, 19syl 17 . . 3 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (𝐽 ∈ Perf ↔ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
2115, 20sylibrd 258 . 2 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o𝐽 ∈ Perf))
22213impia 1114 1 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn ∧ ¬ 𝑋 ≈ 1o) → 𝐽 ∈ Perf)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wne 2930  wral 3051  wrex 3060  c0 4318  {csn 4624   cuni 4903   class class class wbr 5143  cfv 6543  1oc1o 8478  cen 8959  Topctop 22813  Clsdccld 22938  Perfcperf 23057  Frect1 23229  Conncconn 23333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1o 8485  df-en 8963  df-top 22814  df-cld 22941  df-ntr 22942  df-cls 22943  df-lp 23058  df-perf 23059  df-t1 23236  df-conn 23334
This theorem is referenced by: (None)
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