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Theorem kur14lem9 32025
Description: Lemma for kur14 32027. 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 3435 . . . . . 6 𝑠 ∈ V
32elintrab 4788 . . . . 5 (𝑠 {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ↔ ∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥))
4 ssun1 4064 . . . . . . . 8 {𝐴, (𝑋𝐴), (𝐾𝐴)} ⊆ ({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)})
5 ssun1 4064 . . . . . . . . 9 ({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ⊆ (({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))})
6 ssun1 4064 . . . . . . . . . 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 3920 . . . . . . . . 9 (({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ∪ {(𝐾𝐵), 𝐷, (𝐾‘(𝐼𝐴))}) ⊆ 𝑇
95, 8sstri 3893 . . . . . . . 8 ({𝐴, (𝑋𝐴), (𝐾𝐴)} ∪ {𝐵, 𝐶, (𝐼𝐴)}) ⊆ 𝑇
104, 9sstri 3893 . . . . . . 7 {𝐴, (𝑋𝐴), (𝐾𝐴)} ⊆ 𝑇
11 kur14lem.j . . . . . . . . . . 11 𝐽 ∈ Top
12 kur14lem.x . . . . . . . . . . . 12 𝑋 = 𝐽
1312topopn 21186 . . . . . . . . . . 11 (𝐽 ∈ Top → 𝑋𝐽)
1411, 13ax-mp 5 . . . . . . . . . 10 𝑋𝐽
1514elexi 3451 . . . . . . . . 9 𝑋 ∈ V
16 kur14lem.a . . . . . . . . 9 𝐴𝑋
1715, 16ssexi 5110 . . . . . . . 8 𝐴 ∈ V
1817tpid1 4605 . . . . . . 7 𝐴 ∈ {𝐴, (𝑋𝐴), (𝐾𝐴)}
1910, 18sselii 3881 . . . . . 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 32023 . . . . . . . 8 (𝑦𝑇 → (𝑦𝑋 ∧ {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇))
2625simprd 496 . . . . . . 7 (𝑦𝑇 → {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇)
2726rgen 3113 . . . . . 6 𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇
2825simpld 495 . . . . . . . . . 10 (𝑦𝑇𝑦𝑋)
2915elpw2 5132 . . . . . . . . . 10 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
3028, 29sylibr 235 . . . . . . . . 9 (𝑦𝑇𝑦 ∈ 𝒫 𝑋)
3130ssriv 3888 . . . . . . . 8 𝑇 ⊆ 𝒫 𝑋
3215pwex 5165 . . . . . . . . 9 𝒫 𝑋 ∈ V
3332elpw2 5132 . . . . . . . 8 (𝑇 ∈ 𝒫 𝒫 𝑋𝑇 ⊆ 𝒫 𝑋)
3431, 33mpbir 232 . . . . . . 7 𝑇 ∈ 𝒫 𝒫 𝑋
35 eleq2 2869 . . . . . . . . . 10 (𝑥 = 𝑇 → (𝐴𝑥𝐴𝑇))
36 sseq2 3909 . . . . . . . . . . 11 (𝑥 = 𝑇 → ({(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇))
3736raleqbi1dv 3360 . . . . . . . . . 10 (𝑥 = 𝑇 → (∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥 ↔ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇))
3835, 37anbi12d 630 . . . . . . . . 9 (𝑥 = 𝑇 → ((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) ↔ (𝐴𝑇 ∧ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇)))
39 eleq2 2869 . . . . . . . . 9 (𝑥 = 𝑇 → (𝑠𝑥𝑠𝑇))
4038, 39imbi12d 346 . . . . . . . 8 (𝑥 = 𝑇 → (((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥) ↔ ((𝐴𝑇 ∧ ∀𝑦𝑇 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑇) → 𝑠𝑇)))
4140rspccv 3551 . . . . . . 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 694 . . . . 5 (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥) → 𝑠𝑥) → 𝑠𝑇)
443, 43sylbi 218 . . . 4 (𝑠 {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} → 𝑠𝑇)
4544ssriv 3888 . . 3 {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴𝑥 ∧ ∀𝑦𝑥 {(𝑋𝑦), (𝐾𝑦)} ⊆ 𝑥)} ⊆ 𝑇
461, 45eqsstri 3917 . 2 𝑆𝑇
4711, 12, 20, 21, 16, 22, 23, 24, 7kur14lem8 32024 . 2 (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ 14)
48 1nn0 11750 . . 3 1 ∈ ℕ0
49 4nn0 11753 . . 3 4 ∈ ℕ0
5048, 49deccl 11951 . 2 14 ∈ ℕ0
5146, 47, 50hashsslei 13623 1 (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ 14)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1520  wcel 2079  wral 3103  {crab 3107  cdif 3851  cun 3852  wss 3854  𝒫 cpw 4447  {cpr 4468  {ctp 4470   cuni 4739   cint 4776   class class class wbr 4956  cfv 6217  Fincfn 8347  1c1 10373  cle 10511  4c4 11531  cdc 11936  chash 13528  Topctop 21173  intcnt 21297  clsccl 21298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-iin 4822  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-oadd 7948  df-er 8130  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-dju 9165  df-card 9203  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-nn 11476  df-2 11537  df-3 11538  df-4 11539  df-5 11540  df-6 11541  df-7 11542  df-8 11543  df-9 11544  df-n0 11735  df-xnn0 11805  df-z 11819  df-dec 11937  df-uz 12083  df-fz 12732  df-hash 13529  df-top 21174  df-cld 21299  df-ntr 21300  df-cls 21301
This theorem is referenced by:  kur14lem10  32026
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