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Theorem kur14lem9 32521
Description: Lemma for kur14 32523. 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 3483 . . . . . 6 𝑠 ∈ V
32elintrab 4874 . . . . 5 (𝑠 {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ↔ ∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥))
4 ssun1 4134 . . . . . . . 8 {𝐴, (𝑋𝐴), (𝐾𝐴)} ⊆ ({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)})
5 ssun1 4134 . . . . . . . . 9 ({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ⊆ (({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))})
6 ssun1 4134 . . . . . . . . . 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, 7sseqtrri 3990 . . . . . . . . 9 (({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ⊆ 𝑇
95, 8sstri 3962 . . . . . . . 8 ({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ⊆ 𝑇
104, 9sstri 3962 . . . . . . 7 {𝐴, (𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇
11 kur14lem.j . . . . . . . . . . 11 𝐽 ∈ Top
12 kur14lem.x . . . . . . . . . . . 12 𝑋 = 𝐽
1312topopn 21517 . . . . . . . . . . 11 (𝐽 ∈ Top → 𝑋𝐽)
1411, 13ax-mp 5 . . . . . . . . . 10 𝑋𝐽
1514elexi 3499 . . . . . . . . 9 𝑋 ∈ V
16 kur14lem.a . . . . . . . . 9 𝐴𝑋
1715, 16ssexi 5212 . . . . . . . 8 𝐴 ∈ V
1817tpid1 4689 . . . . . . 7 𝐴 ∈ {𝐴, (𝑋𝐴), (𝐾𝐴)}
1910, 18sselii 3950 . . . . . 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 32519 . . . . . . . 8 (𝑦𝑇 → (𝑦𝑋 ∧ {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇))
2625simprd 499 . . . . . . 7 (𝑦𝑇 → {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇)
2726rgen 3143 . . . . . 6 𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇
2825simpld 498 . . . . . . . . . 10 (𝑦𝑇𝑦𝑋)
2915elpw2 5234 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
3028, 29sylibr 237 . . . . . . . . 9 (𝑦𝑇𝑦 ∈ 𝒫 𝑋)
3130ssriv 3957 . . . . . . . 8 𝑇 ⊆ 𝒫 𝑋
3215pwex 5268 . . . . . . . . 9 𝒫 𝑋 ∈ V
3332elpw2 5234 . . . . . . . 8 (𝑇 ∈ 𝒫 𝒫 𝑋𝑇 ⊆ 𝒫 𝑋)
3431, 33mpbir 234 . . . . . . 7 𝑇 ∈ 𝒫 𝒫 𝑋
35 eleq2 2904 . . . . . . . . . 10 (𝑥 = 𝑇 → (𝐴𝑥𝐴𝑇))
36 sseq2 3979 . . . . . . . . . . 11 (𝑥 = 𝑇 → ({(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇))
3736raleqbi1dv 3394 . . . . . . . . . 10 (𝑥 = 𝑇 → (∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇))
3835, 37anbi12d 633 . . . . . . . . 9 (𝑥 = 𝑇 → ((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) ↔ (𝐴𝑇 ∧ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇)))
39 eleq2 2904 . . . . . . . . 9 (𝑥 = 𝑇 → (𝑠𝑥𝑠𝑇))
4038, 39imbi12d 348 . . . . . . . 8 (𝑥 = 𝑇 → (((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥) ↔ ((𝐴𝑇 ∧ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇) → 𝑠𝑇)))
4140rspccv 3606 . . . . . . 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 697 . . . . 5 (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥) → 𝑠𝑇)
443, 43sylbi 220 . . . 4 (𝑠 {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} → 𝑠𝑇)
4544ssriv 3957 . . 3 {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ⊆ 𝑇
461, 45eqsstri 3987 . 2 𝑆𝑇
4711, 12, 20, 21, 16, 22, 23, 24, 7kur14lem8 32520 . 2 (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ 14)
48 1nn0 11910 . . 3 1 ∈ ℕ0
49 4nn0 11913 . . 3 4 ∈ ℕ0
5048, 49deccl 12110 . 2 14 ∈ ℕ0
5146, 47, 50hashsslei 13792 1 (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ 14)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  {crab 3137  cdif 3916  cun 3917  wss 3919  𝒫 cpw 4522  {cpr 4552  {ctp 4554   cuni 4824   cint 4862   class class class wbr 5052  cfv 6343  Fincfn 8505  1c1 10536  cle 10674  4c4 11691  cdc 12095  chash 13695  Topctop 21504  intcnt 21628  clsccl 21629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-iin 4908  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-oadd 8102  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-dju 9327  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-3 11698  df-4 11699  df-5 11700  df-6 11701  df-7 11702  df-8 11703  df-9 11704  df-n0 11895  df-xnn0 11965  df-z 11979  df-dec 12096  df-uz 12241  df-fz 12895  df-hash 13696  df-top 21505  df-cld 21630  df-ntr 21631  df-cls 21632
This theorem is referenced by:  kur14lem10  32522
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