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Theorem kelac2lem 39098
Description: Lemma for kelac2 39099 and dfac21 39100: 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 5185 . . . . 5 {𝑆, {𝒫 𝑆}} ∈ V
2 vex 3411 . . . . . . . 8 𝑥 ∈ V
32elpr 4458 . . . . . . 7 (𝑥 ∈ {𝑆, {𝒫 𝑆}} ↔ (𝑥 = 𝑆𝑥 = {𝒫 𝑆}))
4 vex 3411 . . . . . . . 8 𝑦 ∈ V
54elpr 4458 . . . . . . 7 (𝑦 ∈ {𝑆, {𝒫 𝑆}} ↔ (𝑦 = 𝑆𝑦 = {𝒫 𝑆}))
6 eqtr3 2794 . . . . . . . . 9 ((𝑥 = 𝑆𝑦 = 𝑆) → 𝑥 = 𝑦)
76orcd 860 . . . . . . . 8 ((𝑥 = 𝑆𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
8 ineq12 4065 . . . . . . . . . 10 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = 𝑆) → (𝑥𝑦) = ({𝒫 𝑆} ∩ 𝑆))
9 incom 4060 . . . . . . . . . . 11 ({𝒫 𝑆} ∩ 𝑆) = (𝑆 ∩ {𝒫 𝑆})
10 pwuninel 7742 . . . . . . . . . . . 12 ¬ 𝒫 𝑆𝑆
11 disjsn 4517 . . . . . . . . . . . 12 ((𝑆 ∩ {𝒫 𝑆}) = ∅ ↔ ¬ 𝒫 𝑆𝑆)
1210, 11mpbir 223 . . . . . . . . . . 11 (𝑆 ∩ {𝒫 𝑆}) = ∅
139, 12eqtri 2795 . . . . . . . . . 10 ({𝒫 𝑆} ∩ 𝑆) = ∅
148, 13syl6eq 2823 . . . . . . . . 9 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = 𝑆) → (𝑥𝑦) = ∅)
1514olcd 861 . . . . . . . 8 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = 𝑆) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
16 ineq12 4065 . . . . . . . . . 10 ((𝑥 = 𝑆𝑦 = {𝒫 𝑆}) → (𝑥𝑦) = (𝑆 ∩ {𝒫 𝑆}))
1716, 12syl6eq 2823 . . . . . . . . 9 ((𝑥 = 𝑆𝑦 = {𝒫 𝑆}) → (𝑥𝑦) = ∅)
1817olcd 861 . . . . . . . 8 ((𝑥 = 𝑆𝑦 = {𝒫 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
19 eqtr3 2794 . . . . . . . . 9 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = {𝒫 𝑆}) → 𝑥 = 𝑦)
2019orcd 860 . . . . . . . 8 ((𝑥 = {𝒫 𝑆} ∧ 𝑦 = {𝒫 𝑆}) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
217, 15, 18, 20ccase 1019 . . . . . . 7 (((𝑥 = 𝑆𝑥 = {𝒫 𝑆}) ∧ (𝑦 = 𝑆𝑦 = {𝒫 𝑆})) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
223, 5, 21syl2anb 589 . . . . . 6 ((𝑥 ∈ {𝑆, {𝒫 𝑆}} ∧ 𝑦 ∈ {𝑆, {𝒫 𝑆}}) → (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅))
2322rgen2a 3169 . . . . 5 𝑥 ∈ {𝑆, {𝒫 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 𝑆}} (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)
24 baspartn 21281 . . . . 5 (({𝑆, {𝒫 𝑆}} ∈ V ∧ ∀𝑥 ∈ {𝑆, {𝒫 𝑆}}∀𝑦 ∈ {𝑆, {𝒫 𝑆}} (𝑥 = 𝑦 ∨ (𝑥𝑦) = ∅)) → {𝑆, {𝒫 𝑆}} ∈ TopBases)
251, 23, 24mp2an 680 . . . 4 {𝑆, {𝒫 𝑆}} ∈ TopBases
26 tgcl 21296 . . . 4 ({𝑆, {𝒫 𝑆}} ∈ TopBases → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Top)
2725, 26mp1i 13 . . 3 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Top)
28 prfi 8586 . . . . . 6 {𝑆, {𝒫 𝑆}} ∈ Fin
29 pwfi 8612 . . . . . 6 ({𝑆, {𝒫 𝑆}} ∈ Fin ↔ 𝒫 {𝑆, {𝒫 𝑆}} ∈ Fin)
3028, 29mpbi 222 . . . . 5 𝒫 {𝑆, {𝒫 𝑆}} ∈ Fin
31 tgdom 21305 . . . . . 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 8532 . . . . 5 ((𝒫 {𝑆, {𝒫 𝑆}} ∈ Fin ∧ (topGen‘{𝑆, {𝒫 𝑆}}) ≼ 𝒫 {𝑆, {𝒫 𝑆}}) → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Fin)
3430, 32, 33mp2an 680 . . . 4 (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Fin
3534a1i 11 . . 3 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ Fin)
3627, 35elind 4053 . 2 (𝑆𝑉 → (topGen‘{𝑆, {𝒫 𝑆}}) ∈ (Top ∩ Fin))
37 fincmp 21720 . 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 387  wo 834   = wceq 1508  wcel 2051  wral 3081  Vcvv 3408  cin 3821  c0 4172  𝒫 cpw 4416  {csn 4435  {cpr 4437   cuni 4708   class class class wbr 4925  cfv 6185  cdom 8302  Fincfn 8304  topGenctg 16565  Topctop 21220  TopBasesctb 21272  Compccmp 21713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-int 4746  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-pred 5983  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-wrecs 7748  df-recs 7810  df-rdg 7848  df-1o 7903  df-2o 7904  df-oadd 7907  df-er 8087  df-map 8206  df-en 8305  df-dom 8306  df-sdom 8307  df-fin 8308  df-topgen 16571  df-top 21221  df-bases 21273  df-cmp 21714
This theorem is referenced by:  kelac2  39099  dfac21  39100
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