Theorem kelac2lem 43021

Description: Lemma for kelac2 43022 and dfac21 43023: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)

Assertion

Ref	Expression
kelac2lem	⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp)

Proof of Theorem kelac2lem

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prex 5452	. . . . 5 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ V
2		vex 3492	. . . . . . . 8 ⊢ 𝑥 ∈ V
3	2	elpr 4672	. . . . . . 7 ⊢ (𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}} ↔ (𝑥 = 𝑆 ∨ 𝑥 = {𝒫 ∪ 𝑆}))
4		vex 3492	. . . . . . . 8 ⊢ 𝑦 ∈ V
5	4	elpr 4672	. . . . . . 7 ⊢ (𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}} ↔ (𝑦 = 𝑆 ∨ 𝑦 = {𝒫 ∪ 𝑆}))
6		eqtr3 2766	. . . . . . . . 9 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑆) → 𝑥 = 𝑦)
7	6	orcd 872	. . . . . . . 8 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
8		ineq12 4236	. . . . . . . . . 10 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 ∩ 𝑦) = ({𝒫 ∪ 𝑆} ∩ 𝑆))
9		incom 4230	. . . . . . . . . . 11 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 ∪ 𝑆})
10		pwuninel 8316	. . . . . . . . . . . 12 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
11		disjsn 4736	. . . . . . . . . . . 12 ⊢ ((𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅ ↔ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
12	10, 11	mpbir 231	. . . . . . . . . . 11 ⊢ (𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅
13	9, 12	eqtri 2768	. . . . . . . . . 10 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = ∅
14	8, 13	eqtrdi 2796	. . . . . . . . 9 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 ∩ 𝑦) = ∅)
15	14	olcd 873	. . . . . . . 8 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
16		ineq12 4236	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 ∩ 𝑦) = (𝑆 ∩ {𝒫 ∪ 𝑆}))
17	16, 12	eqtrdi 2796	. . . . . . . . 9 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 ∩ 𝑦) = ∅)
18	17	olcd 873	. . . . . . . 8 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
19		eqtr3 2766	. . . . . . . . 9 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = {𝒫 ∪ 𝑆}) → 𝑥 = 𝑦)
20	19	orcd 872	. . . . . . . 8 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
21	7, 15, 18, 20	ccase 1038	. . . . . . 7 ⊢ (((𝑥 = 𝑆 ∨ 𝑥 = {𝒫 ∪ 𝑆}) ∧ (𝑦 = 𝑆 ∨ 𝑦 = {𝒫 ∪ 𝑆})) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
22	3, 5, 21	syl2anb 597	. . . . . 6 ⊢ ((𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}} ∧ 𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
23	22	rgen2 3205	. . . . 5 ⊢ ∀𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}} (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)
24		baspartn 22982	. . . . 5 ⊢ (({𝑆, {𝒫 ∪ 𝑆}} ∈ V ∧ ∀𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}} (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)) → {𝑆, {𝒫 ∪ 𝑆}} ∈ TopBases)
25	1, 23, 24	mp2an 691	. . . 4 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ TopBases
26		tgcl 22997	. . . 4 ⊢ ({𝑆, {𝒫 ∪ 𝑆}} ∈ TopBases → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Top)
27	25, 26	mp1i 13	. . 3 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Top)
28		prfi 9391	. . . . . 6 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin
29		pwfi 9385	. . . . . 6 ⊢ ({𝑆, {𝒫 ∪ 𝑆}} ∈ Fin ↔ 𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin)
30	28, 29	mpbi 230	. . . . 5 ⊢ 𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin
31		tgdom 23006	. . . . . 6 ⊢ ({𝑆, {𝒫 ∪ 𝑆}} ∈ V → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 ∪ 𝑆}})
32	1, 31	ax-mp 5	. . . . 5 ⊢ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 ∪ 𝑆}}
33		domfi 9255	. . . . 5 ⊢ ((𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin ∧ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 ∪ 𝑆}}) → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin)
34	30, 32, 33	mp2an 691	. . . 4 ⊢ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin
35	34	a1i 11	. . 3 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin)
36	27, 35	elind 4223	. 2 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ (Top ∩ Fin))
37		fincmp 23422	. 2 ⊢ ((topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ (Top ∩ Fin) → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp)
38	36, 37	syl 17	1 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Comp)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ∩ cin 3975  ∅c0 4352  𝒫 cpw 4622  {csn 4648  {cpr 4650  ∪ cuni 4931   class class class wbr 5166  ‘cfv 6573   ≼ cdom 9001  Fincfn 9003  topGenctg 17497  Topctop 22920  TopBasesctb 22973  Compccmp 23415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-om 7904  df-1o 8522  df-2o 8523  df-en 9004  df-dom 9005  df-fin 9007  df-topgen 17503  df-top 22921  df-bases 22974  df-cmp 23416

This theorem is referenced by:  kelac2  43022  dfac21  43023