Theorem kur14lem8 35524

Description: Lemma for kur14 35527. 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 2761	. . 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 2761	. . . 4 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) = ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})
4		hashtplei 14491	. . . 4 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∈ Fin ∧ (♯‘{𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)}) ≤ 3)
5		hashtplei 14491	. . . 4 ⊢ ({𝐵, 𝐶, (𝐼‘𝐴)} ∈ Fin ∧ (♯‘{𝐵, 𝐶, (𝐼‘𝐴)}) ≤ 3)
6		3nn0 12493	. . . 4 ⊢ 3 ∈ ℕ0
7		3p3e6 12363	. . . 4 ⊢ (3 + 3) = 6
8	3, 4, 5, 6, 6, 7	hashunlei 14432	. . 3 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∈ Fin ∧ (♯‘({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})) ≤ 6)
9		hashtplei 14491	. . 3 ⊢ ({(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))} ∈ Fin ∧ (♯‘{(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ≤ 3)
10		6nn0 12496	. . 3 ⊢ 6 ∈ ℕ0
11		6p3e9 12371	. . 3 ⊢ (6 + 3) = 9
12	2, 8, 9, 10, 6, 11	hashunlei 14432	. 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 2761	. . 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 14491	. . 3 ⊢ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∈ Fin ∧ (♯‘{(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))}) ≤ 3)
15		hashprlei 14475	. . 3 ⊢ ({(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))} ∈ Fin ∧ (♯‘{(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}) ≤ 2)
16		2nn0 12492	. . 3 ⊢ 2 ∈ ℕ0
17		3p2e5 12362	. . 3 ⊢ (3 + 2) = 5
18	13, 14, 15, 6, 16, 17	hashunlei 14432	. 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 12499	. 2 ⊢ 9 ∈ ℕ0
20		5nn0 12495	. 2 ⊢ 5 ∈ ℕ0
21		9p5e14 12777	. 2 ⊢ (9 + 5) = ;14
22	1, 12, 18, 19, 20, 21	hashunlei 14432	1 ⊢ (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ ;14)

Syntax hints:   ∧ wa 399   = wceq 1559   ∈ wcel 2141   ∖ cdif 3899   ∪ cun 3900   ⊆ wss 3902  {cpr 4581  {ctp 4583  ∪ cuni 4862   class class class wbr 5097  ‘cfv 6516  Fincfn 8921  1c1 11068   ≤ cle 11211  2c2 12266  3c3 12267  4c4 12268  5c5 12269  6c6 12270  9c9 12273  ;cdc 12682  ♯chash 14337  Topctop 22941  intcnt 23065  clsccl 23066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7713  ax-cnex 11123  ax-resscn 11124  ax-1cn 11125  ax-icn 11126  ax-addcl 11127  ax-addrcl 11128  ax-mulcl 11129  ax-mulrcl 11130  ax-mulcom 11131  ax-addass 11132  ax-mulass 11133  ax-distr 11134  ax-i2m1 11135  ax-1ne0 11136  ax-1rid 11137  ax-rnegex 11138  ax-rrecex 11139  ax-cnre 11140  ax-pre-lttri 11141  ax-pre-lttrn 11142  ax-pre-ltadd 11143  ax-pre-mulgt0 11144

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6283  df-ord 6344  df-on 6345  df-lim 6346  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7348  df-ov 7394  df-oprab 7395  df-mpo 7396  df-om 7842  df-1st 7965  df-2nd 7966  df-frecs 8256  df-wrecs 8287  df-recs 8336  df-rdg 8375  df-1o 8431  df-oadd 8435  df-er 8672  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9853  df-card 9891  df-pnf 11212  df-mnf 11213  df-xr 11214  df-ltxr 11215  df-le 11216  df-sub 11410  df-neg 11411  df-nn 12205  df-2 12274  df-3 12275  df-4 12276  df-5 12277  df-6 12278  df-7 12279  df-8 12280  df-9 12281  df-n0 12476  df-xnn0 12549  df-z 12563  df-dec 12683  df-uz 12834  df-fz 13507  df-hash 14338

This theorem is referenced by:  kur14lem9  35525