Theorem inatsk 10810

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 10722	. . . . . 6 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
2		winaon 10720	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ On)
3		winalim 10727	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inaccw → Lim 𝐴)
4		r1lim 9806	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦))
5	2, 3, 4	syl2anc 582	. . . . . . . . 9 ⊢ (𝐴 ∈ Inaccw → (𝑅1‘𝐴) = ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦))
6	5	eleq2d 2812	. . . . . . . 8 ⊢ (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1‘𝐴) ↔ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦)))
7		eliun 4998	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦))
8	6, 7	bitrdi 286	. . . . . . 7 ⊢ (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1‘𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦)))
9		onelon 6391	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
10	2, 9	sylan 578	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
11		r1pw 9879	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (𝑥 ∈ (𝑅1‘𝑦) ↔ 𝒫 𝑥 ∈ (𝑅1‘suc 𝑦)))
12	10, 11	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (𝑅1‘𝑦) ↔ 𝒫 𝑥 ∈ (𝑅1‘suc 𝑦)))
13		limsuc 7849	. . . . . . . . . . . . 13 ⊢ (Lim 𝐴 → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
14	3, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inaccw → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
15		r1ord2 9815	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → (suc 𝑦 ∈ 𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴)))
16	2, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inaccw → (suc 𝑦 ∈ 𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴)))
17	14, 16	sylbid 239	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inaccw → (𝑦 ∈ 𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴)))
18	17	imp 405	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴))
19	18	sseld 3978	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝒫 𝑥 ∈ (𝑅1‘suc 𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
20	12, 19	sylbid 239	. . . . . . . 8 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (𝑅1‘𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
21	20	rexlimdva 3145	. . . . . . 7 ⊢ (𝐴 ∈ Inaccw → (∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
22	8, 21	sylbid 239	. . . . . 6 ⊢ (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
23	1, 22	syl 17	. . . . 5 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
24	23	imp 405	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝒫 𝑥 ∈ (𝑅1‘𝐴))
25		elssuni 4938	. . . . 5 ⊢ (𝒫 𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ⊆ ∪ (𝑅1‘𝐴))
26		r1tr2 9811	. . . . 5 ⊢ ∪ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴)
27	25, 26	sstrdi 3992	. . . 4 ⊢ (𝒫 𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ⊆ (𝑅1‘𝐴))
28	24, 27	jccil 521	. . 3 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (𝒫 𝑥 ⊆ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
29	28	ralrimiva 3136	. 2 ⊢ (𝐴 ∈ Inacc → ∀𝑥 ∈ (𝑅1‘𝐴)(𝒫 𝑥 ⊆ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
30	1, 2	syl 17	. . . . . . . . 9 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ On)
31		r1suc 9804	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴))
32	31	eleq2d 2812	. . . . . . . . 9 ⊢ (𝐴 ∈ On → (𝑥 ∈ (𝑅1‘suc 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)))
33	30, 32	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ (𝑅1‘suc 𝐴) ↔ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)))
34		rankr1ai 9832	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘suc 𝐴) → (rank‘𝑥) ∈ suc 𝐴)
35	33, 34	biimtrrdi 253	. . . . . . 7 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → (rank‘𝑥) ∈ suc 𝐴))
36	35	imp 405	. . . . . 6 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (rank‘𝑥) ∈ suc 𝐴)
37		fvex 6904	. . . . . . 7 ⊢ (rank‘𝑥) ∈ V
38	37	elsuc 6436	. . . . . 6 ⊢ ((rank‘𝑥) ∈ suc 𝐴 ↔ ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
39	36, 38	sylib 217	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
40	39	orcomd 869	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴))
41		fvex 6904	. . . . . . . 8 ⊢ (𝑅1‘𝐴) ∈ V
42		elpwi 4605	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → 𝑥 ⊆ (𝑅1‘𝐴))
43	42	ad2antlr 725	. . . . . . . 8 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ⊆ (𝑅1‘𝐴))
44		ssdomg 9021	. . . . . . . 8 ⊢ ((𝑅1‘𝐴) ∈ V → (𝑥 ⊆ (𝑅1‘𝐴) → 𝑥 ≼ (𝑅1‘𝐴)))
45	41, 43, 44	mpsyl 68	. . . . . . 7 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≼ (𝑅1‘𝐴))
46		rankcf 10809	. . . . . . . . . 10 ⊢ ¬ 𝑥 ≺ (cf‘(rank‘𝑥))
47		fveq2 6891	. . . . . . . . . . . 12 ⊢ ((rank‘𝑥) = 𝐴 → (cf‘(rank‘𝑥)) = (cf‘𝐴))
48		elina 10719	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
49	48	simp2bi 1143	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
50	47, 49	sylan9eqr 2788	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (cf‘(rank‘𝑥)) = 𝐴)
51	50	breq2d 5156	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (cf‘(rank‘𝑥)) ↔ 𝑥 ≺ 𝐴))
52	46, 51	mtbii 325	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ 𝐴)
53		inar1 10807	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ≈ 𝐴)
54		sdomentr 9139	. . . . . . . . . . . 12 ⊢ ((𝑥 ≺ (𝑅1‘𝐴) ∧ (𝑅1‘𝐴) ≈ 𝐴) → 𝑥 ≺ 𝐴)
55	54	expcom 412	. . . . . . . . . . 11 ⊢ ((𝑅1‘𝐴) ≈ 𝐴 → (𝑥 ≺ (𝑅1‘𝐴) → 𝑥 ≺ 𝐴))
56	53, 55	syl 17	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inacc → (𝑥 ≺ (𝑅1‘𝐴) → 𝑥 ≺ 𝐴))
57	56	adantr 479	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (𝑅1‘𝐴) → 𝑥 ≺ 𝐴))
58	52, 57	mtod 197	. . . . . . . 8 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1‘𝐴))
59	58	adantlr 713	. . . . . . 7 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1‘𝐴))
60		bren2 9004	. . . . . . 7 ⊢ (𝑥 ≈ (𝑅1‘𝐴) ↔ (𝑥 ≼ (𝑅1‘𝐴) ∧ ¬ 𝑥 ≺ (𝑅1‘𝐴)))
61	45, 59, 60	sylanbrc 581	. . . . . 6 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≈ (𝑅1‘𝐴))
62	61	ex 411	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) = 𝐴 → 𝑥 ≈ (𝑅1‘𝐴)))
63		r1elwf 9830	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑅1‘suc 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
64	33, 63	biimtrrdi 253	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On)))
65	64	imp 405	. . . . . . 7 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
66		r1fnon 9801	. . . . . . . . . 10 ⊢ 𝑅1 Fn On
67	66	fndmi 6654	. . . . . . . . 9 ⊢ dom 𝑅1 = On
68	30, 67	eleqtrrdi 2837	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ dom 𝑅1)
69	68	adantr 479	. . . . . . 7 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → 𝐴 ∈ dom 𝑅1)
70		rankr1ag 9836	. . . . . . 7 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
71	65, 69, 70	syl2anc 582	. . . . . 6 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
72	71	biimprd 247	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) ∈ 𝐴 → 𝑥 ∈ (𝑅1‘𝐴)))
73	62, 72	orim12d 962	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴) → (𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴))))
74	40, 73	mpd 15	. . 3 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴)))
75	74	ralrimiva 3136	. 2 ⊢ (𝐴 ∈ Inacc → ∀𝑥 ∈ 𝒫 (𝑅1‘𝐴)(𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴)))
76		eltsk2g 10783	. . 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 581	1 ⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ∈ Tarski)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∃wrex 3060  Vcvv 3463   ⊆ wss 3947  ∅c0 4323  𝒫 cpw 4598  ∪ cuni 4906  ∪ ciun 4994   class class class wbr 5144  dom cdm 5673   “ cima 5676  Oncon0 6366  Lim wlim 6367  suc csuc 6368  ‘cfv 6544   ≈ cen 8961   ≼ cdom 8962   ≺ csdm 8963  𝑅₁cr1 9796  rankcrnk 9797  cfccf 9971  Inacc_wcwina 10714  Inacccina 10715  Tarskictsk 10780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736  ax-inf2 9675  ax-ac2 10495

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3365  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-int 4948  df-iun 4996  df-iin 4997  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-2o 8487  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-oi 9544  df-r1 9798  df-rank 9799  df-card 9973  df-cf 9975  df-acn 9976  df-ac 10150  df-wina 10716  df-ina 10717  df-tsk 10781

This theorem is referenced by:  r1omtsk  10811  r1tskina  10814  grutsk  10854  inagrud  44005