Theorem kur14lem8 30252

Description: Lemma for kur14 30255. 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 2606	. . 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 2606	. . . 4 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) = ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})
4		hashtplei 13067	. . . 4 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∈ Fin ∧ (#‘{𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)}) ≤ 3)
5		hashtplei 13067	. . . 4 ⊢ ({𝐵, 𝐶, (𝐼‘𝐴)} ∈ Fin ∧ (#‘{𝐵, 𝐶, (𝐼‘𝐴)}) ≤ 3)
6		3nn0 11154	. . . 4 ⊢ 3 ∈ ℕ0
7		3p3e6 11005	. . . 4 ⊢ (3 + 3) = 6
8	3, 4, 5, 6, 6, 7	hashunlei 13021	. . 3 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∈ Fin ∧ (#‘({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})) ≤ 6)
9		hashtplei 13067	. . 3 ⊢ ({(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))} ∈ Fin ∧ (#‘{(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ≤ 3)
10		6nn0 11157	. . 3 ⊢ 6 ∈ ℕ0
11		6p3e9 11014	. . 3 ⊢ (6 + 3) = 9
12	2, 8, 9, 10, 6, 11	hashunlei 13021	. 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 2606	. . 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 13067	. . 3 ⊢ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∈ Fin ∧ (#‘{(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))}) ≤ 3)
15		hashprlei 13056	. . 3 ⊢ ({(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))} ∈ Fin ∧ (#‘{(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}) ≤ 2)
16		2nn0 11153	. . 3 ⊢ 2 ∈ ℕ0
17		3p2e5 11004	. . 3 ⊢ (3 + 2) = 5
18	13, 14, 15, 6, 16, 17	hashunlei 13021	. 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 11160	. 2 ⊢ 9 ∈ ℕ0
20		5nn0 11156	. 2 ⊢ 5 ∈ ℕ0
21		9p5e14 11452	. 2 ⊢ (9 + 5) = ;14
22	1, 12, 18, 19, 20, 21	hashunlei 13021	1 ⊢ (𝑇 ∈ Fin ∧ (#‘𝑇) ≤ ;14)

Syntax hints:   ∧ wa 382   = wceq 1474   ∈ wcel 1976   ∖ cdif 3533   ∪ cun 3534   ⊆ wss 3536  {cpr 4123  {ctp 4125  ∪ cuni 4363   class class class wbr 4574  ‘cfv 5787  Fincfn 7815  1c1 9790   ≤ cle 9928  2c2 10914  3c3 10915  4c4 10916  5c5 10917  6c6 10918  9c9 10921  ;cdc 11322  #chash 12931  Topctop 20456  intcnt 20570  clsccl 20571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-rep 4690  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865  ax-pre-mulgt0 9866

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-reu 2899  df-rmo 2900  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-pss 3552  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-tp 4126  df-op 4128  df-uni 4364  df-int 4402  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-tr 4672  df-eprel 4936  df-id 4940  df-po 4946  df-so 4947  df-fr 4984  df-we 4986  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-pred 5580  df-ord 5626  df-on 5627  df-lim 5628  df-suc 5629  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-riota 6486  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-om 6932  df-1st 7033  df-2nd 7034  df-wrecs 7268  df-recs 7329  df-rdg 7367  df-1o 7421  df-oadd 7425  df-er 7603  df-en 7816  df-dom 7817  df-sdom 7818  df-fin 7819  df-card 8622  df-cda 8847  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-sub 10116  df-neg 10117  df-nn 10865  df-2 10923  df-3 10924  df-4 10925  df-5 10926  df-6 10927  df-7 10928  df-8 10929  df-9 10930  df-n0 11137  df-z 11208  df-dec 11323  df-uz 11517  df-fz 12150  df-hash 12932

This theorem is referenced by:  kur14lem9  30253