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Theorem kur14lem9 31322
Description: Lemma for kur14 31324. Since the set 𝑇 is closed under closure and complement, it contains the minimal set 𝑆 as a subset, so 𝑆 also has at most 14 elements. (Indeed 𝑆 = 𝑇, and it's not hard to prove this, but we don't need it for this proof.) (Contributed by Mario Carneiro, 11-Feb-2015.)
Hypotheses
Ref Expression
kur14lem.j 𝐽 ∈ Top
kur14lem.x 𝑋 = 𝐽
kur14lem.k 𝐾 = (cls‘𝐽)
kur14lem.i 𝐼 = (int‘𝐽)
kur14lem.a 𝐴𝑋
kur14lem.b 𝐵 = (𝑋 ∖ (𝐾𝐴))
kur14lem.c 𝐶 = (𝐾‘(𝑋𝐴))
kur14lem.d 𝐷 = (𝐼‘(𝐾𝐴))
kur14lem.t 𝑇 = ((({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ∪ ({(𝐼𝐶), (𝐾𝐷), (𝐼‘(𝐾𝐵))} ∪ {(𝐾‘(𝐼𝐶)), (𝐼‘(𝐾‘(𝐼𝐴)))}))
kur14lem.s 𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}
Assertion
Ref Expression
kur14lem9 (𝑆 ∈ Fin ∧ (#‘𝑆) ≤ 14)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐾   𝑥,𝑦,𝑇   𝑥,𝑋,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐼(𝑥,𝑦)   𝐽(𝑥,𝑦)   𝐾(𝑦)

Proof of Theorem kur14lem9
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 kur14lem.s . . 3 𝑆 = {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)}
2 vex 3234 . . . . . 6 𝑠 ∈ V
32elintrab 4520 . . . . 5 (𝑠 {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ↔ ∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥))
4 ssun1 3809 . . . . . . . 8 {𝐴, (𝑋𝐴), (𝐾𝐴)} ⊆ ({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)})
5 ssun1 3809 . . . . . . . . 9 ({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ⊆ (({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))})
6 ssun1 3809 . . . . . . . . . 10 (({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ⊆ ((({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ∪ ({(𝐼𝐶), (𝐾𝐷), (𝐼‘(𝐾𝐵))} ∪ {(𝐾‘(𝐼𝐶)), (𝐼‘(𝐾‘(𝐼𝐴)))}))
7 kur14lem.t . . . . . . . . . 10 𝑇 = ((({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ∪ ({(𝐼𝐶), (𝐾𝐷), (𝐼‘(𝐾𝐵))} ∪ {(𝐾‘(𝐼𝐶)), (𝐼‘(𝐾‘(𝐼𝐴)))}))
86, 7sseqtr4i 3671 . . . . . . . . 9 (({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ⊆ 𝑇
95, 8sstri 3645 . . . . . . . 8 ({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ⊆ 𝑇
104, 9sstri 3645 . . . . . . 7 {𝐴, (𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇
11 kur14lem.j . . . . . . . . . . 11 𝐽 ∈ Top
12 kur14lem.x . . . . . . . . . . . 12 𝑋 = 𝐽
1312topopn 20759 . . . . . . . . . . 11 (𝐽 ∈ Top → 𝑋𝐽)
1411, 13ax-mp 5 . . . . . . . . . 10 𝑋𝐽
1514elexi 3244 . . . . . . . . 9 𝑋 ∈ V
16 kur14lem.a . . . . . . . . 9 𝐴𝑋
1715, 16ssexi 4836 . . . . . . . 8 𝐴 ∈ V
1817tpid1 4335 . . . . . . 7 𝐴 ∈ {𝐴, (𝑋𝐴), (𝐾𝐴)}
1910, 18sselii 3633 . . . . . 6 𝐴𝑇
20 kur14lem.k . . . . . . . . 9 𝐾 = (cls‘𝐽)
21 kur14lem.i . . . . . . . . 9 𝐼 = (int‘𝐽)
22 kur14lem.b . . . . . . . . 9 𝐵 = (𝑋 ∖ (𝐾𝐴))
23 kur14lem.c . . . . . . . . 9 𝐶 = (𝐾‘(𝑋𝐴))
24 kur14lem.d . . . . . . . . 9 𝐷 = (𝐼‘(𝐾𝐴))
2511, 12, 20, 21, 16, 22, 23, 24, 7kur14lem7 31320 . . . . . . . 8 (𝑦𝑇 → (𝑦𝑋 ∧ {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇))
2625simprd 478 . . . . . . 7 (𝑦𝑇 → {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇)
2726rgen 2951 . . . . . 6 𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇
2825simpld 474 . . . . . . . . . 10 (𝑦𝑇𝑦𝑋)
2915elpw2 4858 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
3028, 29sylibr 224 . . . . . . . . 9 (𝑦𝑇𝑦 ∈ 𝒫 𝑋)
3130ssriv 3640 . . . . . . . 8 𝑇 ⊆ 𝒫 𝑋
3215pwex 4878 . . . . . . . . 9 𝒫 𝑋 ∈ V
3332elpw2 4858 . . . . . . . 8 (𝑇 ∈ 𝒫 𝒫 𝑋𝑇 ⊆ 𝒫 𝑋)
3431, 33mpbir 221 . . . . . . 7 𝑇 ∈ 𝒫 𝒫 𝑋
35 eleq2 2719 . . . . . . . . . 10 (𝑥 = 𝑇 → (𝐴𝑥𝐴𝑇))
36 sseq2 3660 . . . . . . . . . . 11 (𝑥 = 𝑇 → ({(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇))
3736raleqbi1dv 3176 . . . . . . . . . 10 (𝑥 = 𝑇 → (∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇))
3835, 37anbi12d 747 . . . . . . . . 9 (𝑥 = 𝑇 → ((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) ↔ (𝐴𝑇 ∧ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇)))
39 eleq2 2719 . . . . . . . . 9 (𝑥 = 𝑇 → (𝑠𝑥𝑠𝑇))
4038, 39imbi12d 333 . . . . . . . 8 (𝑥 = 𝑇 → (((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥) ↔ ((𝐴𝑇 ∧ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇) → 𝑠𝑇)))
4140rspccv 3337 . . . . . . 7 (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥) → (𝑇 ∈ 𝒫 𝒫 𝑋 → ((𝐴𝑇 ∧ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇) → 𝑠𝑇)))
4234, 41mpi 20 . . . . . 6 (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥) → ((𝐴𝑇 ∧ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇) → 𝑠𝑇))
4319, 27, 42mp2ani 714 . . . . 5 (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥) → 𝑠𝑇)
443, 43sylbi 207 . . . 4 (𝑠 {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} → 𝑠𝑇)
4544ssriv 3640 . . 3 {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ⊆ 𝑇
461, 45eqsstri 3668 . 2 𝑆𝑇
4711, 12, 20, 21, 16, 22, 23, 24, 7kur14lem8 31321 . 2 (𝑇 ∈ Fin ∧ (#‘𝑇) ≤ 14)
48 1nn0 11346 . . 3 1 ∈ ℕ0
49 4nn0 11349 . . 3 4 ∈ ℕ0
5048, 49deccl 11550 . 2 14 ∈ ℕ0
5146, 47, 50hashsslei 13251 1 (𝑆 ∈ Fin ∧ (#‘𝑆) ≤ 14)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  cdif 3604  cun 3605  wss 3607  𝒫 cpw 4191  {cpr 4212  {ctp 4214   cuni 4468   cint 4507   class class class wbr 4685  cfv 5926  Fincfn 7997  1c1 9975  cle 10113  4c4 11110  cdc 11531  #chash 13157  Topctop 20746  intcnt 20869  clsccl 20870
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-xnn0 11402  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-hash 13158  df-top 20747  df-cld 20871  df-ntr 20872  df-cls 20873
This theorem is referenced by:  kur14lem10  31323
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