Theorem kur14lem8 35235

Description: Lemma for kur14 35238. Show that the set 𝑇 contains at most 14 elements. (It could be less if some of the operators take the same value for a given set, but Kuratowski showed that this upper bound of 14 is tight in the sense that there exist topological spaces and subsets of these spaces for which all 14 generated sets are distinct, and indeed the real numbers form such a topological space.) (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	⊢ 𝐴 ⊆ 𝑋
kur14lem.b	⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴))
kur14lem.c	⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴))
kur14lem.d	⊢ 𝐷 = (𝐼‘(𝐾‘𝐴))
kur14lem.t	⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))

Assertion

Ref	Expression
kur14lem8	⊢ (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ ;14)

Proof of Theorem kur14lem8

Step	Hyp	Ref	Expression
1		kur14lem.t	. 2 ⊢ 𝑇 = ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))
2		eqid 2735	. . 3 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) = (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))})
3		eqid 2735	. . . 4 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) = ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})
4		hashtplei 14502	. . . 4 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∈ Fin ∧ (♯‘{𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)}) ≤ 3)
5		hashtplei 14502	. . . 4 ⊢ ({𝐵, 𝐶, (𝐼‘𝐴)} ∈ Fin ∧ (♯‘{𝐵, 𝐶, (𝐼‘𝐴)}) ≤ 3)
6		3nn0 12519	. . . 4 ⊢ 3 ∈ ℕ0
7		3p3e6 12392	. . . 4 ⊢ (3 + 3) = 6
8	3, 4, 5, 6, 6, 7	hashunlei 14443	. . 3 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∈ Fin ∧ (♯‘({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})) ≤ 6)
9		hashtplei 14502	. . 3 ⊢ ({(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))} ∈ Fin ∧ (♯‘{(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ≤ 3)
10		6nn0 12522	. . 3 ⊢ 6 ∈ ℕ0
11		6p3e9 12400	. . 3 ⊢ (6 + 3) = 9
12	2, 8, 9, 10, 6, 11	hashunlei 14443	. 2 ⊢ ((({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ∈ Fin ∧ (♯‘(({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))})) ≤ 9)
13		eqid 2735	. . 3 ⊢ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}) = ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})
14		hashtplei 14502	. . 3 ⊢ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∈ Fin ∧ (♯‘{(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))}) ≤ 3)
15		hashprlei 14486	. . 3 ⊢ ({(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))} ∈ Fin ∧ (♯‘{(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}) ≤ 2)
16		2nn0 12518	. . 3 ⊢ 2 ∈ ℕ0
17		3p2e5 12391	. . 3 ⊢ (3 + 2) = 5
18	13, 14, 15, 6, 16, 17	hashunlei 14443	. 2 ⊢ (({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}) ∈ Fin ∧ (♯‘({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))})) ≤ 5)
19		9nn0 12525	. 2 ⊢ 9 ∈ ℕ0
20		5nn0 12521	. 2 ⊢ 5 ∈ ℕ0
21		9p5e14 12798	. 2 ⊢ (9 + 5) = ;14
22	1, 12, 18, 19, 20, 21	hashunlei 14443	1 ⊢ (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ ;14)

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ∖ cdif 3923   ∪ cun 3924   ⊆ wss 3926  {cpr 4603  {ctp 4605  ∪ cuni 4883   class class class wbr 5119  ‘cfv 6531  Fincfn 8959  1c1 11130   ≤ cle 11270  2c2 12295  3c3 12296  4c4 12297  5c5 12298  6c6 12299  9c9 12302  ;cdc 12708  ♯chash 14348  Topctop 22831  intcnt 22955  clsccl 22956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-1cn 11187  ax-icn 11188  ax-addcl 11189  ax-addrcl 11190  ax-mulcl 11191  ax-mulrcl 11192  ax-mulcom 11193  ax-addass 11194  ax-mulass 11195  ax-distr 11196  ax-i2m1 11197  ax-1ne0 11198  ax-1rid 11199  ax-rnegex 11200  ax-rrecex 11201  ax-cnre 11202  ax-pre-lttri 11203  ax-pre-lttrn 11204  ax-pre-ltadd 11205  ax-pre-mulgt0 11206

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-oadd 8484  df-er 8719  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-dju 9915  df-card 9953  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-sub 11468  df-neg 11469  df-nn 12241  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12502  df-xnn0 12575  df-z 12589  df-dec 12709  df-uz 12853  df-fz 13525  df-hash 14349

This theorem is referenced by:  kur14lem9  35236