Theorem t1connperf 23339

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.

Step	Hyp	Ref	Expression
1		t1connperf.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
2		simplr 768	. . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝐽 ∈ Conn)
3		simprr 772	. . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ 𝐽)
4		vex 3442	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
5	4	snnz 4730	. . . . . . . . 9 ⊢ {𝑥} ≠ ∅
6	5	a1i 11	. . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ≠ ∅)
7	1	t1sncld 23229	. . . . . . . . 9 ⊢ ((𝐽 ∈ Fre ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ (Clsd‘𝐽))
8	7	ad2ant2r 747	. . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ (Clsd‘𝐽))
9	1, 2, 3, 6, 8	connclo 23318	. . . . . . 7 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} = 𝑋)
10	4	ensn1 8953	. . . . . . 7 ⊢ {𝑥} ≈ 1o
11	9, 10	eqbrtrrdi 5135	. . . . . 6 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝑋 ≈ 1o)
12	11	rexlimdvaa 3131	. . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (∃𝑥 ∈ 𝑋 {𝑥} ∈ 𝐽 → 𝑋 ≈ 1o))
13	12	con3d 152	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ¬ ∃𝑥 ∈ 𝑋 {𝑥} ∈ 𝐽))
14		ralnex 3055	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽 ↔ ¬ ∃𝑥 ∈ 𝑋 {𝑥} ∈ 𝐽)
15	13, 14	imbitrrdi 252	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
16		t1top 23233	. . . . 5 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
17	16	adantr 480	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → 𝐽 ∈ Top)
18	1	isperf3 23056	. . . . 5 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
19	18	baib 535	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Perf ↔ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
20	17, 19	syl 17	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (𝐽 ∈ Perf ↔ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
21	15, 20	sylibrd 259	. 2 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → 𝐽 ∈ Perf))
22	21	3impia 1117	1 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn ∧ ¬ 𝑋 ≈ 1o) → 𝐽 ∈ Perf)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  ∅c0 4286  {csn 4579  ∪ cuni 4861   class class class wbr 5095  ‘cfv 6486  1_oc1o 8388   ≈ cen 8876  Topctop 22796  Clsdccld 22919  Perfcperf 23038  Frect1 23210  Conncconn 23314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-int 4900  df-iun 4946  df-iin 4947  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-1o 8395  df-en 8880  df-top 22797  df-cld 22922  df-ntr 22923  df-cls 22924  df-lp 23039  df-perf 23040  df-t1 23217  df-conn 23315

This theorem is referenced by: (None)