MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  t1connperf Structured version   Visualization version   GIF version

Theorem t1connperf 23295
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 766 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝐽 ∈ Conn)
3 simprr 770 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ 𝐽)
4 vex 3472 . . . . . . . . . 10 𝑥 ∈ V
54snnz 4775 . . . . . . . . 9 {𝑥} ≠ ∅
65a1i 11 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ≠ ∅)
71t1sncld 23185 . . . . . . . . 9 ((𝐽 ∈ Fre ∧ 𝑥𝑋) → {𝑥} ∈ (Clsd‘𝐽))
87ad2ant2r 744 . . . . . . . 8 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ (Clsd‘𝐽))
91, 2, 3, 6, 8connclo 23274 . . . . . . 7 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} = 𝑋)
104ensn1 9019 . . . . . . 7 {𝑥} ≈ 1o
119, 10eqbrtrrdi 5181 . . . . . 6 (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝑋 ≈ 1o)
1211rexlimdvaa 3150 . . . . 5 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (∃𝑥𝑋 {𝑥} ∈ 𝐽𝑋 ≈ 1o))
1312con3d 152 . . . 4 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ¬ ∃𝑥𝑋 {𝑥} ∈ 𝐽))
14 ralnex 3066 . . . 4 (∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽 ↔ ¬ ∃𝑥𝑋 {𝑥} ∈ 𝐽)
1513, 14imbitrrdi 251 . . 3 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ∀𝑥𝑋 ¬ {𝑥} ∈ 𝐽))
16 t1top 23189 . . . . 5 (𝐽 ∈ Fre → 𝐽 ∈ Top)
1716adantr 480 . . . 4 ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → 𝐽 ∈ Top)
181isperf3 23012 . . . . 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 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 395  w3a 1084   = wceq 1533  wcel 2098  wne 2934  wral 3055  wrex 3064  c0 4317  {csn 4623   cuni 4902   class class class wbr 5141  cfv 6537  1oc1o 8460  cen 8938  Topctop 22750  Clsdccld 22875  Perfcperf 22994  Frect1 23166  Conncconn 23270
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1o 8467  df-en 8942  df-top 22751  df-cld 22878  df-ntr 22879  df-cls 22880  df-lp 22995  df-perf 22996  df-t1 23173  df-conn 23271
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator