Theorem inatsk 9479

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 9391	. . . . . 6 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
2		winaon 9389	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ On)
3		winalim 9396	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inaccw → Lim 𝐴)
4		r1lim 8518	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦))
5	2, 3, 4	syl2anc 691	. . . . . . . . 9 ⊢ (𝐴 ∈ Inaccw → (𝑅1‘𝐴) = ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦))
6	5	eleq2d 2673	. . . . . . . 8 ⊢ (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1‘𝐴) ↔ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦)))
7		eliun 4460	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦))
8	6, 7	syl6bb 275	. . . . . . 7 ⊢ (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1‘𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦)))
9		onelon 5665	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
10	2, 9	sylan 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
11		r1pw 8591	. . . . . . . . . 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 6941	. . . . . . . . . . . . 13 ⊢ (Lim 𝐴 → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
14	3, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inaccw → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
15		r1ord2 8527	. . . . . . . . . . . . 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 229	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inaccw → (𝑦 ∈ 𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴)))
18	17	imp 444	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴))
19	18	sseld 3567	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝒫 𝑥 ∈ (𝑅1‘suc 𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
20	12, 19	sylbid 229	. . . . . . . 8 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (𝑅1‘𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
21	20	rexlimdva 3013	. . . . . . 7 ⊢ (𝐴 ∈ Inaccw → (∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
22	8, 21	sylbid 229	. . . . . 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 444	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝒫 𝑥 ∈ (𝑅1‘𝐴))
25		elssuni 4403	. . . . 5 ⊢ (𝒫 𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ⊆ ∪ (𝑅1‘𝐴))
26		r1tr2 8523	. . . . 5 ⊢ ∪ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴)
27	25, 26	syl6ss 3580	. . . 4 ⊢ (𝒫 𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ⊆ (𝑅1‘𝐴))
28	24, 27	jccil 561	. . 3 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (𝒫 𝑥 ⊆ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
29	28	ralrimiva 2949	. 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 8516	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴))
32	31	eleq2d 2673	. . . . . . . . 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 8544	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘suc 𝐴) → (rank‘𝑥) ∈ suc 𝐴)
35	33, 34	syl6bir 243	. . . . . . 7 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → (rank‘𝑥) ∈ suc 𝐴))
36	35	imp 444	. . . . . 6 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (rank‘𝑥) ∈ suc 𝐴)
37		fvex 6113	. . . . . . 7 ⊢ (rank‘𝑥) ∈ V
38	37	elsuc 5711	. . . . . 6 ⊢ ((rank‘𝑥) ∈ suc 𝐴 ↔ ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
39	36, 38	sylib 207	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
40	39	orcomd 402	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴))
41		fvex 6113	. . . . . . . 8 ⊢ (𝑅1‘𝐴) ∈ V
42		elpwi 4117	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → 𝑥 ⊆ (𝑅1‘𝐴))
43	42	ad2antlr 759	. . . . . . . 8 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ⊆ (𝑅1‘𝐴))
44		ssdomg 7887	. . . . . . . 8 ⊢ ((𝑅1‘𝐴) ∈ V → (𝑥 ⊆ (𝑅1‘𝐴) → 𝑥 ≼ (𝑅1‘𝐴)))
45	41, 43, 44	mpsyl 66	. . . . . . 7 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≼ (𝑅1‘𝐴))
46		rankcf 9478	. . . . . . . . . 10 ⊢ ¬ 𝑥 ≺ (cf‘(rank‘𝑥))
47		fveq2 6103	. . . . . . . . . . . 12 ⊢ ((rank‘𝑥) = 𝐴 → (cf‘(rank‘𝑥)) = (cf‘𝐴))
48		elina 9388	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
49	48	simp2bi 1070	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
50	47, 49	sylan9eqr 2666	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (cf‘(rank‘𝑥)) = 𝐴)
51	50	breq2d 4595	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (cf‘(rank‘𝑥)) ↔ 𝑥 ≺ 𝐴))
52	46, 51	mtbii 315	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ 𝐴)
53		inar1 9476	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ≈ 𝐴)
54		sdomentr 7979	. . . . . . . . . . . 12 ⊢ ((𝑥 ≺ (𝑅1‘𝐴) ∧ (𝑅1‘𝐴) ≈ 𝐴) → 𝑥 ≺ 𝐴)
55	54	expcom 450	. . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (𝑅1‘𝐴) → 𝑥 ≺ 𝐴))
58	52, 57	mtod 188	. . . . . . . 8 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1‘𝐴))
59	58	adantlr 747	. . . . . . 7 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1‘𝐴))
60		bren2 7872	. . . . . . 7 ⊢ (𝑥 ≈ (𝑅1‘𝐴) ↔ (𝑥 ≼ (𝑅1‘𝐴) ∧ ¬ 𝑥 ≺ (𝑅1‘𝐴)))
61	45, 59, 60	sylanbrc 695	. . . . . 6 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≈ (𝑅1‘𝐴))
62	61	ex 449	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) = 𝐴 → 𝑥 ≈ (𝑅1‘𝐴)))
63		r1elwf 8542	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑅1‘suc 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
64	33, 63	syl6bir 243	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On)))
65	64	imp 444	. . . . . . 7 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
66		r1fnon 8513	. . . . . . . . . 10 ⊢ 𝑅1 Fn On
67		fndm 5904	. . . . . . . . . 10 ⊢ (𝑅1 Fn On → dom 𝑅1 = On)
68	66, 67	ax-mp 5	. . . . . . . . 9 ⊢ dom 𝑅1 = On
69	30, 68	syl6eleqr 2699	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ dom 𝑅1)
70	69	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → 𝐴 ∈ dom 𝑅1)
71		rankr1ag 8548	. . . . . . 7 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
72	65, 70, 71	syl2anc 691	. . . . . 6 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
73	72	biimprd 237	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) ∈ 𝐴 → 𝑥 ∈ (𝑅1‘𝐴)))
74	62, 73	orim12d 879	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴) → (𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴))))
75	40, 74	mpd 15	. . 3 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴)))
76	75	ralrimiva 2949	. 2 ⊢ (𝐴 ∈ Inacc → ∀𝑥 ∈ 𝒫 (𝑅1‘𝐴)(𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴)))
77		eltsk2g 9452	. . 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)))))
78	41, 77	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))))
79	29, 76, 78	sylanbrc 695	1 ⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ∈ Tarski)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583  dom cdm 5038   “ cima 5041  Oncon0 5640  Lim wlim 5641  suc csuc 5642   Fn wfn 5799  ‘cfv 5804   ≈ cen 7838   ≼ cdom 7839   ≺ csdm 7840  𝑅₁cr1 8508  rankcrnk 8509  cfccf 8646  Inacc_wcwina 9383  Inacccina 9384  Tarskictsk 9449

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-r1 8510  df-rank 8511  df-card 8648  df-cf 8650  df-acn 8651  df-ac 8822  df-wina 9385  df-ina 9386  df-tsk 9450

This theorem is referenced by:  r1omtsk  9480  r1tskina  9483  grutsk  9523