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Theorem t1connperf 23445
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 768 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝐽 ∈ Conn)
3 simprr 772 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ 𝐽)
4 vex 3483 . . . . . . . . . 10 𝑥 ∈ V
54snnz 4775 . . . . . . . . 9 {𝑥} ≠ ∅
65a1i 11 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ≠ ∅)
71t1sncld 23335 . . . . . . . . 9 ((𝐽 ∈ Fre ∧ 𝑥𝑋) → {𝑥} ∈ (Clsd‘𝐽))
87ad2ant2r 747 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ (Clsd‘𝐽))
91, 2, 3, 6, 8connclo 23424 . . . . . . 7 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} = 𝑋)
104ensn1 9062 . . . . . . 7 {𝑥} ≈ 1o
119, 10eqbrtrrdi 5182 . . . . . 6 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝑋 ≈ 1o)
1211rexlimdvaa 3155 . . . . 5 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (∃𝑥𝑋 {𝑥} ∈ 𝐽𝑋 ≈ 1o))
1312con3d 152 . . . 4 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ¬ ∃𝑥𝑋 {𝑥} ∈ 𝐽))
14 ralnex 3071 . . . 4 (∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽 ↔ ¬ ∃𝑥𝑋 {𝑥} ∈ 𝐽)
1513, 14imbitrrdi 252 . . 3 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
16 t1top 23339 . . . . 5 (𝐽 ∈ Fre → 𝐽 ∈ Top)
1716adantr 480 . . . 4 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → 𝐽 ∈ Top)
181isperf3 23162 . . . . 5 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
1918baib 535 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Perf ↔ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
2017, 19syl 17 . . 3 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (𝐽 ∈ Perf ↔ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
2115, 20sylibrd 259 . 2 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o𝐽 ∈ Perf))
22213impia 1117 1 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn ∧ ¬ 𝑋 ≈ 1o) → 𝐽 ∈ Perf)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  wrex 3069  c0 4332  {csn 4625   cuni 4906   class class class wbr 5142  cfv 6560  1oc1o 8500  cen 8983  Topctop 22900  Clsdccld 23025  Perfcperf 23144  Frect1 23316  Conncconn 23420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-1o 8507  df-en 8987  df-top 22901  df-cld 23028  df-ntr 23029  df-cls 23030  df-lp 23145  df-perf 23146  df-t1 23323  df-conn 23421
This theorem is referenced by: (None)
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