Theorem kelac2lem 40889

Description: Lemma for kelac2 40890 and dfac21 40891: 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 5355	. . . . 5 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ V
2		vex 3436	. . . . . . . 8 ⊢ 𝑥 ∈ V
3	2	elpr 4584	. . . . . . 7 ⊢ (𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}} ↔ (𝑥 = 𝑆 ∨ 𝑥 = {𝒫 ∪ 𝑆}))
4		vex 3436	. . . . . . . 8 ⊢ 𝑦 ∈ V
5	4	elpr 4584	. . . . . . 7 ⊢ (𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}} ↔ (𝑦 = 𝑆 ∨ 𝑦 = {𝒫 ∪ 𝑆}))
6		eqtr3 2764	. . . . . . . . 9 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑆) → 𝑥 = 𝑦)
7	6	orcd 870	. . . . . . . 8 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
8		ineq12 4141	. . . . . . . . . 10 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 ∩ 𝑦) = ({𝒫 ∪ 𝑆} ∩ 𝑆))
9		incom 4135	. . . . . . . . . . 11 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 ∪ 𝑆})
10		pwuninel 8091	. . . . . . . . . . . 12 ⊢ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆
11		disjsn 4647	. . . . . . . . . . . 12 ⊢ ((𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅ ↔ ¬ 𝒫 ∪ 𝑆 ∈ 𝑆)
12	10, 11	mpbir 230	. . . . . . . . . . 11 ⊢ (𝑆 ∩ {𝒫 ∪ 𝑆}) = ∅
13	9, 12	eqtri 2766	. . . . . . . . . 10 ⊢ ({𝒫 ∪ 𝑆} ∩ 𝑆) = ∅
14	8, 13	eqtrdi 2794	. . . . . . . . 9 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 ∩ 𝑦) = ∅)
15	14	olcd 871	. . . . . . . 8 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
16		ineq12 4141	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 ∩ 𝑦) = (𝑆 ∩ {𝒫 ∪ 𝑆}))
17	16, 12	eqtrdi 2794	. . . . . . . . 9 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 ∩ 𝑦) = ∅)
18	17	olcd 871	. . . . . . . 8 ⊢ ((𝑥 = 𝑆 ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
19		eqtr3 2764	. . . . . . . . 9 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = {𝒫 ∪ 𝑆}) → 𝑥 = 𝑦)
20	19	orcd 870	. . . . . . . 8 ⊢ ((𝑥 = {𝒫 ∪ 𝑆} ∧ 𝑦 = {𝒫 ∪ 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
21	7, 15, 18, 20	ccase 1035	. . . . . . 7 ⊢ (((𝑥 = 𝑆 ∨ 𝑥 = {𝒫 ∪ 𝑆}) ∧ (𝑦 = 𝑆 ∨ 𝑦 = {𝒫 ∪ 𝑆})) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
22	3, 5, 21	syl2anb 598	. . . . . 6 ⊢ ((𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}} ∧ 𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}}) → (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅))
23	22	rgen2 3120	. . . . 5 ⊢ ∀𝑥 ∈ {𝑆, {𝒫 ∪ 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 ∪ 𝑆}} (𝑥 = 𝑦 ∨ (𝑥 ∩ 𝑦) = ∅)
24		baspartn 22104	. . . . 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 689	. . . 4 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ TopBases
26		tgcl 22119	. . . 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 9089	. . . . . 6 ⊢ {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin
29		pwfi 8961	. . . . . 6 ⊢ ({𝑆, {𝒫 ∪ 𝑆}} ∈ Fin ↔ 𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin)
30	28, 29	mpbi 229	. . . . 5 ⊢ 𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin
31		tgdom 22128	. . . . . 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 8975	. . . . 5 ⊢ ((𝒫 {𝑆, {𝒫 ∪ 𝑆}} ∈ Fin ∧ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 ∪ 𝑆}}) → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin)
34	30, 32, 33	mp2an 689	. . . 4 ⊢ (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin
35	34	a1i 11	. . 3 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ Fin)
36	27, 35	elind 4128	. 2 ⊢ (𝑆 ∈ 𝑉 → (topGen‘{𝑆, {𝒫 ∪ 𝑆}}) ∈ (Top ∩ Fin))
37		fincmp 22544	. 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 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432   ∩ cin 3886  ∅c0 4256  𝒫 cpw 4533  {csn 4561  {cpr 4563  ∪ cuni 4839   class class class wbr 5074  ‘cfv 6433   ≼ cdom 8731  Fincfn 8733  topGenctg 17148  Topctop 22042  TopBasesctb 22095  Compccmp 22537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-en 8734  df-dom 8735  df-fin 8737  df-topgen 17154  df-top 22043  df-bases 22096  df-cmp 22538

This theorem is referenced by:  kelac2  40890  dfac21  40891