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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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