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Theorem kelac2lem 42627
Description: Lemma for kelac2 42628 and dfac21 42629: knob topologies are compact. (Contributed by Stefan O'Rear, 22-Feb-2015.)
Assertion
Ref Expression
kelac2lem (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Comp)

Proof of Theorem kelac2lem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prex 5434 . . . . 5 {𝑆, {𝒫 𝑆}} ∈ V
2 vex 3465 . . . . . . . 8 𝑥 ∈ V
32elpr 4654 . . . . . . 7 (𝑥 ∈ {𝑆, {𝒫 𝑆}} ↔ (𝑥 = 𝑆𝑥 = {𝒫 𝑆}))
4 vex 3465 . . . . . . . 8 𝑦 ∈ V
54elpr 4654 . . . . . . 7 (𝑦 ∈ {𝑆, {𝒫 𝑆}} ↔ (𝑦 = 𝑆𝑦 = {𝒫 𝑆}))
6 eqtr3 2751 . . . . . . . . 9 ((𝑥 = 𝑆𝑦 = 𝑆) → 𝑥 = 𝑦)
76orcd 871 . . . . . . . 8 ((𝑥 = 𝑆𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
8 ineq12 4205 . . . . . . . . . 10 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = 𝑆) → (𝑥𝑦) = ({𝒫 𝑆} ∩ 𝑆))
9 incom 4199 . . . . . . . . . . 11 ({𝒫 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 𝑆})
10 pwuninel 8281 . . . . . . . . . . . 12 ¬ 𝒫 𝑆𝑆
11 disjsn 4717 . . . . . . . . . . . 12 ((𝑆 ∩ {𝒫 𝑆}) = ∅ ↔ ¬ 𝒫 𝑆𝑆)
1210, 11mpbir 230 . . . . . . . . . . 11 (𝑆 ∩ {𝒫 𝑆}) = ∅
139, 12eqtri 2753 . . . . . . . . . 10 ({𝒫 𝑆} ∩ 𝑆) = ∅
148, 13eqtrdi 2781 . . . . . . . . 9 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = 𝑆) → (𝑥𝑦) = ∅)
1514olcd 872 . . . . . . . 8 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
16 ineq12 4205 . . . . . . . . . 10 ((𝑥 = 𝑆𝑦 = {𝒫 𝑆}) → (𝑥𝑦) = (𝑆 ∩ {𝒫 𝑆}))
1716, 12eqtrdi 2781 . . . . . . . . 9 ((𝑥 = 𝑆𝑦 = {𝒫 𝑆}) → (𝑥𝑦) = ∅)
1817olcd 872 . . . . . . . 8 ((𝑥 = 𝑆𝑦 = {𝒫 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
19 eqtr3 2751 . . . . . . . . 9 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = {𝒫 𝑆}) → 𝑥 = 𝑦)
2019orcd 871 . . . . . . . 8 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = {𝒫 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
217, 15, 18, 20ccase 1035 . . . . . . 7 (((𝑥 = 𝑆𝑥 = {𝒫 𝑆}) ∧ (𝑦 = 𝑆𝑦 = {𝒫 𝑆})) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
223, 5, 21syl2anb 596 . . . . . 6 ((𝑥 ∈ {𝑆, {𝒫 𝑆}} ∧ 𝑦 ∈ {𝑆, {𝒫 𝑆}}) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
2322rgen2 3187 . . . . 5 𝑥 ∈ {𝑆, {𝒫 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 𝑆}} (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)
24 baspartn 22901 . . . . 5 (({𝑆, {𝒫 𝑆}} ∈ V ∧ ∀𝑥 ∈ {𝑆, {𝒫 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 𝑆}} (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → {𝑆, {𝒫 𝑆}} ∈ TopBases)
251, 23, 24mp2an 690 . . . 4 {𝑆, {𝒫 𝑆}} ∈ TopBases
26 tgcl 22916 . . . 4 ({𝑆, {𝒫 𝑆}} ∈ TopBases → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Top)
2725, 26mp1i 13 . . 3 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Top)
28 prfi 9348 . . . . . 6 {𝑆, {𝒫 𝑆}} ∈ Fin
29 pwfi 9203 . . . . . 6 ({𝑆, {𝒫 𝑆}} ∈ Fin ↔ 𝒫 {𝑆, {𝒫 𝑆}} ∈ Fin)
3028, 29mpbi 229 . . . . 5 𝒫 {𝑆, {𝒫 𝑆}} ∈ Fin
31 tgdom 22925 . . . . . 6 ({𝑆, {𝒫 𝑆}} ∈ V → (topGen‘{𝑆, {𝒫 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 𝑆}})
321, 31ax-mp 5 . . . . 5 (topGen‘{𝑆, {𝒫 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 𝑆}}
33 domfi 9217 . . . . 5 ((𝒫 {𝑆, {𝒫 𝑆}} ∈ Fin ∧ (topGen‘{𝑆, {𝒫 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 𝑆}}) → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Fin)
3430, 32, 33mp2an 690 . . . 4 (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Fin
3534a1i 11 . . 3 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Fin)
3627, 35elind 4192 . 2 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ (Top ∩ Fin))
37 fincmp 23341 . 2 ((topGen‘{𝑆, {𝒫 𝑆}}) ∈ (Top ∩ Fin) → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Comp)
3836, 37syl 17 1 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394  wo 845   = wceq 1533  wcel 2098  wral 3050  Vcvv 3461  cin 3943  c0 4322  𝒫 cpw 4604  {csn 4630  {cpr 4632   cuni 4909   class class class wbr 5149  cfv 6549  cdom 8962  Fincfn 8964  topGenctg 17422  Topctop 22839  TopBasesctb 22892  Compccmp 23334
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 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  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 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-1o 8487  df-en 8965  df-dom 8966  df-fin 8968  df-topgen 17428  df-top 22840  df-bases 22893  df-cmp 23335
This theorem is referenced by:  kelac2  42628  dfac21  42629
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