Theorem kur14lem8 35245

Description: Lemma for kur14 35248. 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 2731	. . 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 2731	. . . 4 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) = ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})
4		hashtplei 14388	. . . 4 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∈ Fin ∧ (♯‘{𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)}) ≤ 3)
5		hashtplei 14388	. . . 4 ⊢ ({𝐵, 𝐶, (𝐼‘𝐴)} ∈ Fin ∧ (♯‘{𝐵, 𝐶, (𝐼‘𝐴)}) ≤ 3)
6		3nn0 12396	. . . 4 ⊢ 3 ∈ ℕ0
7		3p3e6 12269	. . . 4 ⊢ (3 + 3) = 6
8	3, 4, 5, 6, 6, 7	hashunlei 14329	. . 3 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∈ Fin ∧ (♯‘({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})) ≤ 6)
9		hashtplei 14388	. . 3 ⊢ ({(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))} ∈ Fin ∧ (♯‘{(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ≤ 3)
10		6nn0 12399	. . 3 ⊢ 6 ∈ ℕ0
11		6p3e9 12277	. . 3 ⊢ (6 + 3) = 9
12	2, 8, 9, 10, 6, 11	hashunlei 14329	. 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 2731	. . 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 14388	. . 3 ⊢ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∈ Fin ∧ (♯‘{(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))}) ≤ 3)
15		hashprlei 14372	. . 3 ⊢ ({(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))} ∈ Fin ∧ (♯‘{(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}) ≤ 2)
16		2nn0 12395	. . 3 ⊢ 2 ∈ ℕ0
17		3p2e5 12268	. . 3 ⊢ (3 + 2) = 5
18	13, 14, 15, 6, 16, 17	hashunlei 14329	. 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 12402	. 2 ⊢ 9 ∈ ℕ0
20		5nn0 12398	. 2 ⊢ 5 ∈ ℕ0
21		9p5e14 12675	. 2 ⊢ (9 + 5) = ;14
22	1, 12, 18, 19, 20, 21	hashunlei 14329	1 ⊢ (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ ;14)

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2111   ∖ cdif 3899   ∪ cun 3900   ⊆ wss 3902  {cpr 4578  {ctp 4580  ∪ cuni 4859   class class class wbr 5091  ‘cfv 6481  Fincfn 8869  1c1 11004   ≤ cle 11144  2c2 12177  3c3 12178  4c4 12179  5c5 12180  6c6 12181  9c9 12184  ;cdc 12585  ♯chash 14234  Topctop 22806  intcnt 22930  clsccl 22931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060  ax-1cn 11061  ax-icn 11062  ax-addcl 11063  ax-addrcl 11064  ax-mulcl 11065  ax-mulrcl 11066  ax-mulcom 11067  ax-addass 11068  ax-mulass 11069  ax-distr 11070  ax-i2m1 11071  ax-1ne0 11072  ax-1rid 11073  ax-rnegex 11074  ax-rrecex 11075  ax-cnre 11076  ax-pre-lttri 11077  ax-pre-lttrn 11078  ax-pre-ltadd 11079  ax-pre-mulgt0 11080

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-dju 9791  df-card 9829  df-pnf 11145  df-mnf 11146  df-xr 11147  df-ltxr 11148  df-le 11149  df-sub 11343  df-neg 11344  df-nn 12123  df-2 12185  df-3 12186  df-4 12187  df-5 12188  df-6 12189  df-7 12190  df-8 12191  df-9 12192  df-n0 12379  df-xnn0 12452  df-z 12466  df-dec 12586  df-uz 12730  df-fz 13405  df-hash 14235

This theorem is referenced by:  kur14lem9  35246