Theorem kur14lem9 35208

Description: Lemma for kur14 35210. 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 4927	. . . . 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 4144	. . . . . . . 8 ⊢ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})
5		ssun1 4144	. . . . . . . . 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 4144	. . . . . . . . . 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 3999	. . . . . . . . 9 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ⊆ 𝑇
9	5, 8	sstri 3959	. . . . . . . 8 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ⊆ 𝑇
10	4, 9	sstri 3959	. . . . . . 7 ⊢ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇
11		kur14lem.j	. . . . . . . . . . 11 ⊢ 𝐽 ∈ Top
12		kur14lem.x	. . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝐽
13	12	topopn 22800	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
14	11, 13	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑋 ∈ 𝐽
15	14	elexi 3473	. . . . . . . . 9 ⊢ 𝑋 ∈ V
16		kur14lem.a	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑋
17	15, 16	ssexi 5280	. . . . . . . 8 ⊢ 𝐴 ∈ V
18	17	tpid1 4735	. . . . . . 7 ⊢ 𝐴 ∈ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)}
19	10, 18	sselii 3946	. . . . . 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 35206	. . . . . . . 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 3047	. . . . . 6 ⊢ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇
28	25	simpld 494	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑇 → 𝑦 ⊆ 𝑋)
29	15	elpw2 5292	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
30	28, 29	sylibr 234	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑇 → 𝑦 ∈ 𝒫 𝑋)
31	30	ssriv 3953	. . . . . . . 8 ⊢ 𝑇 ⊆ 𝒫 𝑋
32	15	pwex 5338	. . . . . . . . 9 ⊢ 𝒫 𝑋 ∈ V
33	32	elpw2 5292	. . . . . . . 8 ⊢ (𝑇 ∈ 𝒫 𝒫 𝑋 ↔ 𝑇 ⊆ 𝒫 𝑋)
34	31, 33	mpbir 231	. . . . . . 7 ⊢ 𝑇 ∈ 𝒫 𝒫 𝑋
35		eleq2 2818	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑇))
36		sseq2 3976	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → ({(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇))
37	36	raleqbi1dv 3313	. . . . . . . . . 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 2818	. . . . . . . . 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 3588	. . . . . . 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 3953	. . 3 ⊢ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ⊆ 𝑇
46	1, 45	eqsstri 3996	. 2 ⊢ 𝑆 ⊆ 𝑇
47	11, 12, 20, 21, 16, 22, 23, 24, 7	kur14lem8 35207	. 2 ⊢ (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ ;14)
48		1nn0 12465	. . 3 ⊢ 1 ∈ ℕ0
49		4nn0 12468	. . 3 ⊢ 4 ∈ ℕ0
50	48, 49	deccl 12671	. 2 ⊢ ;14 ∈ ℕ0
51	46, 47, 50	hashsslei 14398	1 ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408   ∖ cdif 3914   ∪ cun 3915   ⊆ wss 3917  𝒫 cpw 4566  {cpr 4594  {ctp 4596  ∪ cuni 4874  ∩ cint 4913   class class class wbr 5110  ‘cfv 6514  Fincfn 8921  1c1 11076   ≤ cle 11216  4c4 12250  ;cdc 12656  ♯chash 14302  Topctop 22787  intcnt 22911  clsccl 22912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-iin 4961  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-oadd 8441  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-dju 9861  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-4 12258  df-5 12259  df-6 12260  df-7 12261  df-8 12262  df-9 12263  df-n0 12450  df-xnn0 12523  df-z 12537  df-dec 12657  df-uz 12801  df-fz 13476  df-hash 14303  df-top 22788  df-cld 22913  df-ntr 22914  df-cls 22915

This theorem is referenced by:  kur14lem10  35209