Theorem kur14lem9 35198

Description: Lemma for kur14 35200. 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	⊢ 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)))}))
kur14lem.s	⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 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.

Step	Hyp	Ref	Expression
1		kur14lem.s	. . 3 ⊢ 𝑆 = ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)}
2		vex 3481	. . . . . 6 ⊢ 𝑠 ∈ V
3	2	elintrab 4964	. . . . 5 ⊢ (𝑠 ∈ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ↔ ∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) → 𝑠 ∈ 𝑥))
4		ssun1 4187	. . . . . . . 8 ⊢ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})
5		ssun1 4187	. . . . . . . . 9 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ⊆ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))})
6		ssun1 4187	. . . . . . . . . 10 ⊢ (({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))}) ∪ ({(𝐼‘𝐶), (𝐾‘𝐷), (𝐼‘(𝐾‘𝐵))} ∪ {(𝐾‘(𝐼‘𝐶)), (𝐼‘(𝐾‘(𝐼‘𝐴)))}))
7		kur14lem.t	. . . . . . . . . 10 ⊢ 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)))}))
8	6, 7	sseqtrri 4032	. . . . . . . . 9 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ⊆ 𝑇
9	5, 8	sstri 4004	. . . . . . . 8 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ⊆ 𝑇
10	4, 9	sstri 4004	. . . . . . 7 ⊢ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇
11		kur14lem.j	. . . . . . . . . . 11 ⊢ 𝐽 ∈ Top
12		kur14lem.x	. . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝐽
13	12	topopn 22927	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
14	11, 13	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑋 ∈ 𝐽
15	14	elexi 3500	. . . . . . . . 9 ⊢ 𝑋 ∈ V
16		kur14lem.a	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑋
17	15, 16	ssexi 5327	. . . . . . . 8 ⊢ 𝐴 ∈ V
18	17	tpid1 4772	. . . . . . 7 ⊢ 𝐴 ∈ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)}
19	10, 18	sselii 3991	. . . . . 6 ⊢ 𝐴 ∈ 𝑇
20		kur14lem.k	. . . . . . . . 9 ⊢ 𝐾 = (cls‘𝐽)
21		kur14lem.i	. . . . . . . . 9 ⊢ 𝐼 = (int‘𝐽)
22		kur14lem.b	. . . . . . . . 9 ⊢ 𝐵 = (𝑋 ∖ (𝐾‘𝐴))
23		kur14lem.c	. . . . . . . . 9 ⊢ 𝐶 = (𝐾‘(𝑋 ∖ 𝐴))
24		kur14lem.d	. . . . . . . . 9 ⊢ 𝐷 = (𝐼‘(𝐾‘𝐴))
25	11, 12, 20, 21, 16, 22, 23, 24, 7	kur14lem7 35196	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑇 → (𝑦 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇))
26	25	simprd 495	. . . . . . 7 ⊢ (𝑦 ∈ 𝑇 → {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇)
27	26	rgen 3060	. . . . . 6 ⊢ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇
28	25	simpld 494	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑇 → 𝑦 ⊆ 𝑋)
29	15	elpw2 5339	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
30	28, 29	sylibr 234	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑇 → 𝑦 ∈ 𝒫 𝑋)
31	30	ssriv 3998	. . . . . . . 8 ⊢ 𝑇 ⊆ 𝒫 𝑋
32	15	pwex 5385	. . . . . . . . 9 ⊢ 𝒫 𝑋 ∈ V
33	32	elpw2 5339	. . . . . . . 8 ⊢ (𝑇 ∈ 𝒫 𝒫 𝑋 ↔ 𝑇 ⊆ 𝒫 𝑋)
34	31, 33	mpbir 231	. . . . . . 7 ⊢ 𝑇 ∈ 𝒫 𝒫 𝑋
35		eleq2 2827	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑇))
36		sseq2 4021	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → ({(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇))
37	36	raleqbi1dv 3335	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → (∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇))
38	35, 37	anbi12d 632	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → ((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) ↔ (𝐴 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇)))
39		eleq2 2827	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → (𝑠 ∈ 𝑥 ↔ 𝑠 ∈ 𝑇))
40	38, 39	imbi12d 344	. . . . . . . 8 ⊢ (𝑥 = 𝑇 → (((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) → 𝑠 ∈ 𝑥) ↔ ((𝐴 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇) → 𝑠 ∈ 𝑇)))
41	40	rspccv 3618	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) → 𝑠 ∈ 𝑥) → (𝑇 ∈ 𝒫 𝒫 𝑋 → ((𝐴 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇) → 𝑠 ∈ 𝑇)))
42	34, 41	mpi 20	. . . . . 6 ⊢ (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) → 𝑠 ∈ 𝑥) → ((𝐴 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇) → 𝑠 ∈ 𝑇))
43	19, 27, 42	mp2ani 698	. . . . 5 ⊢ (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) → 𝑠 ∈ 𝑥) → 𝑠 ∈ 𝑇)
44	3, 43	sylbi 217	. . . 4 ⊢ (𝑠 ∈ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} → 𝑠 ∈ 𝑇)
45	44	ssriv 3998	. . 3 ⊢ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ⊆ 𝑇
46	1, 45	eqsstri 4029	. 2 ⊢ 𝑆 ⊆ 𝑇
47	11, 12, 20, 21, 16, 22, 23, 24, 7	kur14lem8 35197	. 2 ⊢ (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ ;14)
48		1nn0 12539	. . 3 ⊢ 1 ∈ ℕ0
49		4nn0 12542	. . 3 ⊢ 4 ∈ ℕ0
50	48, 49	deccl 12745	. 2 ⊢ ;14 ∈ ℕ0
51	46, 47, 50	hashsslei 14461	1 ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058  {crab 3432   ∖ cdif 3959   ∪ cun 3960   ⊆ wss 3962  𝒫 cpw 4604  {cpr 4632  {ctp 4634  ∪ cuni 4911  ∩ cint 4950   class class class wbr 5147  ‘cfv 6562  Fincfn 8983  1c1 11153   ≤ cle 11293  4c4 12320  ;cdc 12730  ♯chash 14365  Topctop 22914  intcnt 23040  clsccl 23041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-xnn0 12597  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-hash 14366  df-top 22915  df-cld 23042  df-ntr 23043  df-cls 23044

This theorem is referenced by:  kur14lem10  35199