Theorem kur14lem9 35796

Description: Lemma for kur14 35798. 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 3454	. . . . . 6 ⊢ 𝑠 ∈ V
3	2	elintrab 4920	. . . . 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 4124	. . . . . . . 8 ⊢ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})
5		ssun1 4124	. . . . . . . . 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 4124	. . . . . . . . . 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 3980	. . . . . . . . 9 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ⊆ 𝑇
9	5, 8	sstri 3940	. . . . . . . 8 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ⊆ 𝑇
10	4, 9	sstri 3940	. . . . . . 7 ⊢ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇
11		kur14lem.j	. . . . . . . . . . 11 ⊢ 𝐽 ∈ Top
12		kur14lem.x	. . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝐽
13	12	topopn 23134	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
14	11, 13	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑋 ∈ 𝐽
15	14	elexi 3472	. . . . . . . . 9 ⊢ 𝑋 ∈ V
16		kur14lem.a	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑋
17	15, 16	ssexi 5287	. . . . . . . 8 ⊢ 𝐴 ∈ V
18	17	tpid1 4729	. . . . . . 7 ⊢ 𝐴 ∈ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)}
19	10, 18	sselii 3928	. . . . . 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 35794	. . . . . . . 8 ⊢ (𝑦 ∈ 𝑇 → (𝑦 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇))
26	25	simprd 501	. . . . . . 7 ⊢ (𝑦 ∈ 𝑇 → {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇)
27	26	rgen 3078	. . . . . 6 ⊢ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇
28	25	simpld 500	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑇 → 𝑦 ⊆ 𝑋)
29	15	elpw2 5299	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
30	28, 29	sylibr 237	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑇 → 𝑦 ∈ 𝒫 𝑋)
31	30	ssriv 3935	. . . . . . . 8 ⊢ 𝑇 ⊆ 𝒫 𝑋
32	15	pwex 5345	. . . . . . . . 9 ⊢ 𝒫 𝑋 ∈ V
33	32	elpw2 5299	. . . . . . . 8 ⊢ (𝑇 ∈ 𝒫 𝒫 𝑋 ↔ 𝑇 ⊆ 𝒫 𝑋)
34	31, 33	mpbir 234	. . . . . . 7 ⊢ 𝑇 ∈ 𝒫 𝒫 𝑋
35		eleq2 2849	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑇))
36		sseq2 3957	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → ({(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇))
37	36	raleqbi1dv 3329	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → (∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇))
38	35, 37	anbi12d 644	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → ((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) ↔ (𝐴 ∈ 𝑇 ∧ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇)))
39		eleq2 2849	. . . . . . . . 9 ⊢ (𝑥 = 𝑇 → (𝑠 ∈ 𝑥 ↔ 𝑠 ∈ 𝑇))
40	38, 39	imbi12d 347	. . . . . . . 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 3573	. . . . . . 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 21	. . . . . 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 711	. . . . 5 ⊢ (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) → 𝑠 ∈ 𝑥) → 𝑠 ∈ 𝑇)
44	3, 43	sylbi 220	. . . 4 ⊢ (𝑠 ∈ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} → 𝑠 ∈ 𝑇)
45	44	ssriv 3935	. . 3 ⊢ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ⊆ 𝑇
46	1, 45	eqsstri 3977	. 2 ⊢ 𝑆 ⊆ 𝑇
47	11, 12, 20, 21, 16, 22, 23, 24, 7	kur14lem8 35795	. 2 ⊢ (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ ;14)
48		1nn0 12547	. . 3 ⊢ 1 ∈ ℕ0
49		4nn0 12550	. . 3 ⊢ 4 ∈ ℕ0
50	48, 49	deccl 12754	. 2 ⊢ ;14 ∈ ℕ0
51	46, 47, 50	hashsslei 14494	1 ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  𝒫 cpw 4557  {cpr 4586  {ctp 4588  ∪ cuni 4867  ∩ cint 4907   class class class wbr 5103  ‘cfv 6533  Fincfn 8955  1c1 11128   ≤ cle 11271  4c4 12324  ;cdc 12739  ♯chash 14397  Topctop 23121  intcnt 23245  clsccl 23246

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-addrcl 11188  ax-mulcl 11189  ax-mulrcl 11190  ax-mulcom 11191  ax-addass 11192  ax-mulass 11193  ax-distr 11194  ax-i2m1 11195  ax-1ne0 11196  ax-1rid 11197  ax-rnegex 11198  ax-rrecex 11199  ax-cnre 11200  ax-pre-lttri 11201  ax-pre-lttrn 11202  ax-pre-ltadd 11203  ax-pre-mulgt0 11204

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-oadd 8462  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-dju 9909  df-card 9947  df-pnf 11272  df-mnf 11273  df-xr 11274  df-ltxr 11275  df-le 11276  df-sub 11470  df-neg 11471  df-nn 12261  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12532  df-xnn0 12605  df-z 12619  df-dec 12740  df-uz 12891  df-fz 13565  df-hash 14398  df-top 23122  df-cld 23247  df-ntr 23248  df-cls 23249

This theorem is used by:  kur14lem10  35797