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Theorem kelac2lem 36652
 Description: Lemma for kelac2 36653 and dfac21 36654: 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 4836 . . . . 5 {𝑆, {𝒫 𝑆}} ∈ V
2 vex 3176 . . . . . . . 8 𝑥 ∈ V
32elpr 4146 . . . . . . 7 (𝑥 ∈ {𝑆, {𝒫 𝑆}} ↔ (𝑥 = 𝑆𝑥 = {𝒫 𝑆}))
4 vex 3176 . . . . . . . 8 𝑦 ∈ V
54elpr 4146 . . . . . . 7 (𝑦 ∈ {𝑆, {𝒫 𝑆}} ↔ (𝑦 = 𝑆𝑦 = {𝒫 𝑆}))
6 eqtr3 2631 . . . . . . . . 9 ((𝑥 = 𝑆𝑦 = 𝑆) → 𝑥 = 𝑦)
76orcd 406 . . . . . . . 8 ((𝑥 = 𝑆𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
8 ineq12 3771 . . . . . . . . . 10 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = 𝑆) → (𝑥𝑦) = ({𝒫 𝑆} ∩ 𝑆))
9 incom 3767 . . . . . . . . . . 11 ({𝒫 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 𝑆})
10 pwuninel 7288 . . . . . . . . . . . 12 ¬ 𝒫 𝑆𝑆
11 disjsn 4192 . . . . . . . . . . . 12 ((𝑆 ∩ {𝒫 𝑆}) = ∅ ↔ ¬ 𝒫 𝑆𝑆)
1210, 11mpbir 220 . . . . . . . . . . 11 (𝑆 ∩ {𝒫 𝑆}) = ∅
139, 12eqtri 2632 . . . . . . . . . 10 ({𝒫 𝑆} ∩ 𝑆) = ∅
148, 13syl6eq 2660 . . . . . . . . 9 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = 𝑆) → (𝑥𝑦) = ∅)
1514olcd 407 . . . . . . . 8 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
16 ineq12 3771 . . . . . . . . . 10 ((𝑥 = 𝑆𝑦 = {𝒫 𝑆}) → (𝑥𝑦) = (𝑆 ∩ {𝒫 𝑆}))
1716, 12syl6eq 2660 . . . . . . . . 9 ((𝑥 = 𝑆𝑦 = {𝒫 𝑆}) → (𝑥𝑦) = ∅)
1817olcd 407 . . . . . . . 8 ((𝑥 = 𝑆𝑦 = {𝒫 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
19 eqtr3 2631 . . . . . . . . 9 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = {𝒫 𝑆}) → 𝑥 = 𝑦)
2019orcd 406 . . . . . . . 8 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = {𝒫 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
217, 15, 18, 20ccase 984 . . . . . . 7 (((𝑥 = 𝑆𝑥 = {𝒫 𝑆}) ∧ (𝑦 = 𝑆𝑦 = {𝒫 𝑆})) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
223, 5, 21syl2anb 495 . . . . . 6 ((𝑥 ∈ {𝑆, {𝒫 𝑆}} ∧ 𝑦 ∈ {𝑆, {𝒫 𝑆}}) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
2322rgen2a 2960 . . . . 5 𝑥 ∈ {𝑆, {𝒫 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 𝑆}} (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)
24 baspartn 20569 . . . . 5 (({𝑆, {𝒫 𝑆}} ∈ V ∧ ∀𝑥 ∈ {𝑆, {𝒫 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 𝑆}} (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → {𝑆, {𝒫 𝑆}} ∈ TopBases)
251, 23, 24mp2an 704 . . . 4 {𝑆, {𝒫 𝑆}} ∈ TopBases
26 tgcl 20584 . . . 4 ({𝑆, {𝒫 𝑆}} ∈ TopBases → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Top)
2725, 26mp1i 13 . . 3 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Top)
28 prfi 8120 . . . . . 6 {𝑆, {𝒫 𝑆}} ∈ Fin
29 pwfi 8144 . . . . . 6 ({𝑆, {𝒫 𝑆}} ∈ Fin ↔ 𝒫 {𝑆, {𝒫 𝑆}} ∈ Fin)
3028, 29mpbi 219 . . . . 5 𝒫 {𝑆, {𝒫 𝑆}} ∈ Fin
31 tgdom 20593 . . . . . 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 8066 . . . . 5 ((𝒫 {𝑆, {𝒫 𝑆}} ∈ Fin ∧ (topGen‘{𝑆, {𝒫 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 𝑆}}) → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Fin)
3430, 32, 33mp2an 704 . . . 4 (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Fin
3534a1i 11 . . 3 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Fin)
3627, 35elind 3760 . 2 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ (Top ∩ Fin))
37 fincmp 21006 . 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   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∩ cin 3539  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  ∪ cuni 4372   class class class wbr 4583  ‘cfv 5804   ≼ cdom 7839  Fincfn 7841  topGenctg 15921  Topctop 20517  TopBasesctb 20520  Compccmp 20999 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-topgen 15927  df-top 20521  df-bases 20522  df-cmp 21000 This theorem is referenced by:  kelac2  36653  dfac21  36654
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