Theorem t1connperf 23572

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 780	. . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝐽 ∈ Conn)
3		simprr 784	. . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ 𝐽)
4		vex 3457	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
5	4	snnz 4741	. . . . . . . . 9 ⊢ {𝑥} ≠ ∅
6	5	a1i 11	. . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ≠ ∅)
7	1	t1sncld 23462	. . . . . . . . 9 ⊢ ((𝐽 ∈ Fre ∧ 𝑥 ∈ 𝑋) → {𝑥} ∈ (Clsd‘𝐽))
8	7	ad2ant2r 759	. . . . . . . 8 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} ∈ (Clsd‘𝐽))
9	1, 2, 3, 6, 8	connclo 23551	. . . . . . 7 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → {𝑥} = 𝑋)
10	4	ensn1 9017	. . . . . . 7 ⊢ {𝑥} ≈ 1o
11	9, 10	eqbrtrrdi 5150	. . . . . 6 ⊢ (((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) ∧ (𝑥 ∈ 𝑋 ∧ {𝑥} ∈ 𝐽)) → 𝑋 ≈ 1o)
12	11	rexlimdvaa 3165	. . . . 5 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (∃𝑥 ∈ 𝑋 {𝑥} ∈ 𝐽 → 𝑋 ≈ 1o))
13	12	con3d 153	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ¬ ∃𝑥 ∈ 𝑋 {𝑥} ∈ 𝐽))
14		ralnex 3089	. . . 4 ⊢ (∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽 ↔ ¬ ∃𝑥 ∈ 𝑋 {𝑥} ∈ 𝐽)
15	13, 14	imbitrrdi 255	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
16		t1top 23466	. . . . 5 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top)
17	16	adantr 485	. . . 4 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → 𝐽 ∈ Top)
18	1	isperf3 23289	. . . . 5 ⊢ (𝐽 ∈ Perf ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
19	18	baib 544	. . . 4 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Perf ↔ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
20	17, 19	syl 18	. . 3 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (𝐽 ∈ Perf ↔ ∀𝑥 ∈ 𝑋 ¬ {𝑥} ∈ 𝐽))
21	15, 20	sylibrd 262	. 2 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn) → (¬ 𝑋 ≈ 1o → 𝐽 ∈ Perf))
22	21	3impia 1133	1 ⊢ ((𝐽 ∈ Fre ∧ 𝐽 ∈ Conn ∧ ¬ 𝑋 ≈ 1o) → 𝐽 ∈ Perf)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  ∅c0 4285  {csn 4588  ∪ cuni 4871   class class class wbr 5108  ‘cfv 6536  1_oc1o 8445   ≈ cen 8939  Topctop 23029  Clsdccld 23152  Perfcperf 23271  Frect1 23443  Conncconn 23547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-iin 4958  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-1o 8452  df-en 8943  df-top 23030  df-cld 23155  df-ntr 23156  df-cls 23157  df-lp 23272  df-perf 23273  df-t1 23450  df-conn 23548

This theorem is referenced by: (None)