Theorem inatsk 10763

Description: (𝑅₁‘𝐴) for 𝐴 a strongly inaccessible cardinal is a Tarski class. (Contributed by Mario Carneiro, 8-Jun-2013.)

Assertion

Ref	Expression
inatsk	⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ∈ Tarski)

Proof of Theorem inatsk

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inawina 10675	. . . . . 6 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
2		winaon 10673	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ On)
3		winalim 10680	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inaccw → Lim 𝐴)
4		r1lim 9744	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦))
5	2, 3, 4	syl2anc 595	. . . . . . . . 9 ⊢ (𝐴 ∈ Inaccw → (𝑅1‘𝐴) = ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦))
6	5	eleq2d 2855	. . . . . . . 8 ⊢ (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1‘𝐴) ↔ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦)))
7		eliun 4964	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦))
8	6, 7	bitrdi 290	. . . . . . 7 ⊢ (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1‘𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦)))
9		onelon 6386	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
10	2, 9	sylan 591	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
11		r1pw 9817	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (𝑥 ∈ (𝑅1‘𝑦) ↔ 𝒫 𝑥 ∈ (𝑅1‘suc 𝑦)))
12	10, 11	syl 18	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (𝑅1‘𝑦) ↔ 𝒫 𝑥 ∈ (𝑅1‘suc 𝑦)))
13		limsuc 7845	. . . . . . . . . . . . 13 ⊢ (Lim 𝐴 → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
14	3, 13	syl 18	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inaccw → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
15		r1ord2 9753	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → (suc 𝑦 ∈ 𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴)))
16	2, 15	syl 18	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inaccw → (suc 𝑦 ∈ 𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴)))
17	14, 16	sylbid 243	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inaccw → (𝑦 ∈ 𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴)))
18	17	imp 411	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴))
19	18	sseld 3944	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝒫 𝑥 ∈ (𝑅1‘suc 𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
20	12, 19	sylbid 243	. . . . . . . 8 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (𝑅1‘𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
21	20	rexlimdva 3172	. . . . . . 7 ⊢ (𝐴 ∈ Inaccw → (∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
22	8, 21	sylbid 243	. . . . . 6 ⊢ (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
23	1, 22	syl 18	. . . . 5 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
24	23	imp 411	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝒫 𝑥 ∈ (𝑅1‘𝐴))
25		elssuni 4908	. . . . 5 ⊢ (𝒫 𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ⊆ ∪ (𝑅1‘𝐴))
26		r1tr2 9749	. . . . 5 ⊢ ∪ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴)
27	25, 26	sstrdi 3957	. . . 4 ⊢ (𝒫 𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ⊆ (𝑅1‘𝐴))
28	24, 27	jccil 531	. . 3 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (𝒫 𝑥 ⊆ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
29	28	ralrimiva 3163	. 2 ⊢ (𝐴 ∈ Inacc → ∀𝑥 ∈ (𝑅1‘𝐴)(𝒫 𝑥 ⊆ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
30	1, 2	syl 18	. . . . . . . . 9 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ On)
31		r1suc 9742	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴))
32	31	eleq2d 2855	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝑥 ∈ (𝑅1‘suc 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)))
33	30, 32	syl 18	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ (𝑅1‘suc 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)))
34		rankr1ai 9770	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘suc 𝐴) → (rank‘𝑥) ∈ suc 𝐴)
35	33, 34	biimtrrdi 257	. . . . . . 7 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → (rank‘𝑥) ∈ suc 𝐴))
36	35	imp 411	. . . . . 6 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (rank‘𝑥) ∈ suc 𝐴)
37		fvex 6895	. . . . . . 7 ⊢ (rank‘𝑥) ∈ V
38	37	elsuc 6434	. . . . . 6 ⊢ ((rank‘𝑥) ∈ suc 𝐴 ↔ ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
39	36, 38	sylib 221	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
40	39	orcomd 884	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴))
41		fvex 6895	. . . . . . . 8 ⊢ (𝑅1‘𝐴) ∈ V
42		elpwi 4574	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → 𝑥 ⊆ (𝑅1‘𝐴))
43	42	ad2antlr 739	. . . . . . . 8 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ⊆ (𝑅1‘𝐴))
44		ssdomg 8997	. . . . . . . 8 ⊢ ((𝑅1‘𝐴) ∈ V → (𝑥 ⊆ (𝑅1‘𝐴) → 𝑥 ≼ (𝑅1‘𝐴)))
45	41, 43, 44	mpsyl 69	. . . . . . 7 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≼ (𝑅1‘𝐴))
46		rankcf 10762	. . . . . . . . . 10 ⊢ ¬ 𝑥 ≺ (cf‘(rank‘𝑥))
47		fveq2 6882	. . . . . . . . . . . 12 ⊢ ((rank‘𝑥) = 𝐴 → (cf‘(rank‘𝑥)) = (cf‘𝐴))
48		elina 10672	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
49	48	simp2bi 1162	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
50	47, 49	sylan9eqr 2826	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (cf‘(rank‘𝑥)) = 𝐴)
51	50	breq2d 5125	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (cf‘(rank‘𝑥)) ↔ 𝑥 ≺ 𝐴))
52	46, 51	mtbii 329	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ 𝐴)
53		inar1 10760	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ≈ 𝐴)
54		sdomentr 9099	. . . . . . . . . . . 12 ⊢ ((𝑥 ≺ (𝑅1‘𝐴) ∧ (𝑅1‘𝐴) ≈ 𝐴) → 𝑥 ≺ 𝐴)
55	54	expcom 418	. . . . . . . . . . 11 ⊢ ((𝑅1‘𝐴) ≈ 𝐴 → (𝑥 ≺ (𝑅1‘𝐴) → 𝑥 ≺ 𝐴))
56	53, 55	syl 18	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inacc → (𝑥 ≺ (𝑅1‘𝐴) → 𝑥 ≺ 𝐴))
57	56	adantr 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (𝑅1‘𝐴) → 𝑥 ≺ 𝐴))
58	52, 57	mtod 201	. . . . . . . 8 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1‘𝐴))
59	58	adantlr 727	. . . . . . 7 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1‘𝐴))
60		bren2 8980	. . . . . . 7 ⊢ (𝑥 ≈ (𝑅1‘𝐴) ↔ (𝑥 ≼ (𝑅1‘𝐴) ∧ ¬ 𝑥 ≺ (𝑅1‘𝐴)))
61	45, 59, 60	sylanbrc 594	. . . . . 6 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≈ (𝑅1‘𝐴))
62	61	ex 417	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) = 𝐴 → 𝑥 ≈ (𝑅1‘𝐴)))
63		r1elwf 9768	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑅1‘suc 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
64	33, 63	biimtrrdi 257	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On)))
65	64	imp 411	. . . . . . 7 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
66		r1fnon 9739	. . . . . . . . . 10 ⊢ 𝑅1 Fn On
67	66	fndmi 6640	. . . . . . . . 9 ⊢ dom 𝑅1 = On
68	30, 67	eleqtrrdi 2880	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ dom 𝑅1)
69	68	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → 𝐴 ∈ dom 𝑅1)
70		rankr1ag 9774	. . . . . . 7 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
71	65, 69, 70	syl2anc 595	. . . . . 6 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
72	71	biimprd 251	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) ∈ 𝐴 → 𝑥 ∈ (𝑅1‘𝐴)))
73	62, 72	orim12d 979	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴) → (𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴))))
74	40, 73	mpd 16	. . 3 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴)))
75	74	ralrimiva 3163	. 2 ⊢ (𝐴 ∈ Inacc → ∀𝑥 ∈ 𝒫 (𝑅1‘𝐴)(𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴)))
76		eltsk2g 10736	. . 3 ⊢ ((𝑅1‘𝐴) ∈ V → ((𝑅1‘𝐴) ∈ Tarski ↔ (∀𝑥 ∈ (𝑅1‘𝐴)(𝒫 𝑥 ⊆ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴)) ∧ ∀𝑥 ∈ 𝒫 (𝑅1‘𝐴)(𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴)))))
77	41, 76	ax-mp 5	. 2 ⊢ ((𝑅1‘𝐴) ∈ Tarski ↔ (∀𝑥 ∈ (𝑅1‘𝐴)(𝒫 𝑥 ⊆ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴)) ∧ ∀𝑥 ∈ 𝒫 (𝑅1‘𝐴)(𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴))))
78	29, 75, 77	sylanbrc 594	1 ⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ∈ Tarski)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  ∪ cuni 4876  ∪ ciun 4960   class class class wbr 5113  dom cdm 5662   “ cima 5665  Oncon0 6361  Lim wlim 6362  suc csuc 6363  ‘cfv 6537   ≈ cen 8940   ≼ cdom 8941   ≺ csdm 8942  𝑅₁cr1 9734  rankcrnk 9735  cfccf 9923  Inacc_wcwina 10667  Inacccina 10668  Tarskictsk 10733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9610  ax-ac2 10447

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-oi 9472  df-r1 9736  df-rank 9737  df-card 9925  df-cf 9927  df-acn 9928  df-ac 10100  df-wina 10669  df-ina 10670  df-tsk 10734

This theorem is referenced by:  r1omtsk  10764  r1tskina  10767  grutsk  10807  inagrud  44898