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Theorem t1connperf 21748
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 756 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝐽 ∈ Conn)
3 simprr 760 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ 𝐽)
4 vex 3418 . . . . . . . . . 10 𝑥 ∈ V
54snnz 4585 . . . . . . . . 9 {𝑥} ≠ ∅
65a1i 11 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ≠ ∅)
71t1sncld 21638 . . . . . . . . 9 ((𝐽 ∈ Fre ∧ 𝑥𝑋) → {𝑥} ∈ (Clsd‘𝐽))
87ad2ant2r 734 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ (Clsd‘𝐽))
91, 2, 3, 6, 8connclo 21727 . . . . . . 7 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} = 𝑋)
104ensn1 8370 . . . . . . 7 {𝑥} ≈ 1o
119, 10syl6eqbrr 4969 . . . . . 6 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝑋 ≈ 1o)
1211rexlimdvaa 3230 . . . . 5 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (∃𝑥𝑋 {𝑥} ∈ 𝐽𝑋 ≈ 1o))
1312con3d 150 . . . 4 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ¬ ∃𝑥𝑋 {𝑥} ∈ 𝐽))
14 ralnex 3183 . . . 4 (∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽 ↔ ¬ ∃𝑥𝑋 {𝑥} ∈ 𝐽)
1513, 14syl6ibr 244 . . 3 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
16 t1top 21642 . . . . 5 (𝐽 ∈ Fre → 𝐽 ∈ Top)
1716adantr 473 . . . 4 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → 𝐽 ∈ Top)
181isperf3 21465 . . . . 5 (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
1918baib 528 . . . 4 (𝐽 ∈ Top → (𝐽 ∈ Perf ↔ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
2017, 19syl 17 . . 3 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (𝐽 ∈ Perf ↔ ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
2115, 20sylibrd 251 . 2 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o𝐽 ∈ Perf))
22213impia 1097 1 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn ∧ ¬ 𝑋 ≈ 1o) → 𝐽 ∈ Perf)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  wne 2967  wral 3088  wrex 3089  c0 4178  {csn 4441   cuni 4712   class class class wbr 4929  cfv 6188  1oc1o 7898  cen 8303  Topctop 21205  Clsdccld 21328  Perfcperf 21447  Frect1 21619  Conncconn 21723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-int 4750  df-iun 4794  df-iin 4795  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-suc 6035  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-1o 7905  df-en 8307  df-top 21206  df-cld 21331  df-ntr 21332  df-cls 21333  df-lp 21448  df-perf 21449  df-t1 21626  df-conn 21724
This theorem is referenced by: (None)
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