Theorem kur14lem9 33076

Description: Lemma for kur14 33078. 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 3426	. . . . . 6 ⊢ 𝑠 ∈ V
3	2	elintrab 4888	. . . . 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 4102	. . . . . . . 8 ⊢ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)})
5		ssun1 4102	. . . . . . . . 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 4102	. . . . . . . . . 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 3954	. . . . . . . . 9 ⊢ (({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ∪ {(𝐾‘𝐵), 𝐷, (𝐾‘(𝐼‘𝐴))}) ⊆ 𝑇
9	5, 8	sstri 3926	. . . . . . . 8 ⊢ ({𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ∪ {𝐵, 𝐶, (𝐼‘𝐴)}) ⊆ 𝑇
10	4, 9	sstri 3926	. . . . . . 7 ⊢ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇
11		kur14lem.j	. . . . . . . . . . 11 ⊢ 𝐽 ∈ Top
12		kur14lem.x	. . . . . . . . . . . 12 ⊢ 𝑋 = ∪ 𝐽
13	12	topopn 21963	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
14	11, 13	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑋 ∈ 𝐽
15	14	elexi 3441	. . . . . . . . 9 ⊢ 𝑋 ∈ V
16		kur14lem.a	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝑋
17	15, 16	ssexi 5241	. . . . . . . 8 ⊢ 𝐴 ∈ V
18	17	tpid1 4701	. . . . . . 7 ⊢ 𝐴 ∈ {𝐴, (𝑋 ∖ 𝐴), (𝐾‘𝐴)}
19	10, 18	sselii 3914	. . . . . 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 33074	. . . . . . . 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 3073	. . . . . 6 ⊢ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇
28	25	simpld 494	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑇 → 𝑦 ⊆ 𝑋)
29	15	elpw2 5264	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝒫 𝑋 ↔ 𝑦 ⊆ 𝑋)
30	28, 29	sylibr 233	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑇 → 𝑦 ∈ 𝒫 𝑋)
31	30	ssriv 3921	. . . . . . . 8 ⊢ 𝑇 ⊆ 𝒫 𝑋
32	15	pwex 5298	. . . . . . . . 9 ⊢ 𝒫 𝑋 ∈ V
33	32	elpw2 5264	. . . . . . . 8 ⊢ (𝑇 ∈ 𝒫 𝒫 𝑋 ↔ 𝑇 ⊆ 𝒫 𝑋)
34	31, 33	mpbir 230	. . . . . . 7 ⊢ 𝑇 ∈ 𝒫 𝒫 𝑋
35		eleq2 2827	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑇))
36		sseq2 3943	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑇 → ({(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇))
37	36	raleqbi1dv 3331	. . . . . . . . . 10 ⊢ (𝑥 = 𝑇 → (∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑇 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑇))
38	35, 37	anbi12d 630	. . . . . . . . 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 3549	. . . . . . 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 694	. . . . 5 ⊢ (∀𝑥 ∈ 𝒫 𝒫 𝑋((𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥) → 𝑠 ∈ 𝑥) → 𝑠 ∈ 𝑇)
44	3, 43	sylbi 216	. . . 4 ⊢ (𝑠 ∈ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} → 𝑠 ∈ 𝑇)
45	44	ssriv 3921	. . 3 ⊢ ∩ {𝑥 ∈ 𝒫 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 {(𝑋 ∖ 𝑦), (𝐾‘𝑦)} ⊆ 𝑥)} ⊆ 𝑇
46	1, 45	eqsstri 3951	. 2 ⊢ 𝑆 ⊆ 𝑇
47	11, 12, 20, 21, 16, 22, 23, 24, 7	kur14lem8 33075	. 2 ⊢ (𝑇 ∈ Fin ∧ (♯‘𝑇) ≤ ;14)
48		1nn0 12179	. . 3 ⊢ 1 ∈ ℕ0
49		4nn0 12182	. . 3 ⊢ 4 ∈ ℕ0
50	48, 49	deccl 12381	. 2 ⊢ ;14 ∈ ℕ0
51	46, 47, 50	hashsslei 14069	1 ⊢ (𝑆 ∈ Fin ∧ (♯‘𝑆) ≤ ;14)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067   ∖ cdif 3880   ∪ cun 3881   ⊆ wss 3883  𝒫 cpw 4530  {cpr 4560  {ctp 4562  ∪ cuni 4836  ∩ cint 4876   class class class wbr 5070  ‘cfv 6418  Fincfn 8691  1c1 10803   ≤ cle 10941  4c4 11960  ;cdc 12366  ♯chash 13972  Topctop 21950  intcnt 22076  clsccl 22077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-oadd 8271  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-dju 9590  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-xnn0 12236  df-z 12250  df-dec 12367  df-uz 12512  df-fz 13169  df-hash 13973  df-top 21951  df-cld 22078  df-ntr 22079  df-cls 22080

This theorem is referenced by:  kur14lem10  33077