Theorem inatsk 10736

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 10648	. . . . . 6 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
2		winaon 10646	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ On)
3		winalim 10653	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inaccw → Lim 𝐴)
4		r1lim 9730	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦))
5	2, 3, 4	syl2anc 593	. . . . . . . . 9 ⊢ (𝐴 ∈ Inaccw → (𝑅1‘𝐴) = ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦))
6	5	eleq2d 2848	. . . . . . . 8 ⊢ (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1‘𝐴) ↔ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦)))
7		eliun 4953	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦))
8	6, 7	bitrdi 289	. . . . . . 7 ⊢ (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1‘𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦)))
9		onelon 6371	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
10	2, 9	sylan 589	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
11		r1pw 9803	. . . . . . . . . 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 7829	. . . . . . . . . . . . 13 ⊢ (Lim 𝐴 → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
14	3, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inaccw → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
15		r1ord2 9739	. . . . . . . . . . . . 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 242	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inaccw → (𝑦 ∈ 𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴)))
18	17	imp 410	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴))
19	18	sseld 3935	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝒫 𝑥 ∈ (𝑅1‘suc 𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
20	12, 19	sylbid 242	. . . . . . . 8 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (𝑅1‘𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
21	20	rexlimdva 3163	. . . . . . 7 ⊢ (𝐴 ∈ Inaccw → (∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
22	8, 21	sylbid 242	. . . . . 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 410	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝒫 𝑥 ∈ (𝑅1‘𝐴))
25		elssuni 4897	. . . . 5 ⊢ (𝒫 𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ⊆ ∪ (𝑅1‘𝐴))
26		r1tr2 9735	. . . . 5 ⊢ ∪ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴)
27	25, 26	sstrdi 3948	. . . 4 ⊢ (𝒫 𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ⊆ (𝑅1‘𝐴))
28	24, 27	jccil 530	. . 3 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (𝒫 𝑥 ⊆ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
29	28	ralrimiva 3154	. 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 9728	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴))
32	31	eleq2d 2848	. . . . . . . . 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 9756	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘suc 𝐴) → (rank‘𝑥) ∈ suc 𝐴)
35	33, 34	biimtrrdi 256	. . . . . . 7 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → (rank‘𝑥) ∈ suc 𝐴))
36	35	imp 410	. . . . . 6 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (rank‘𝑥) ∈ suc 𝐴)
37		fvex 6880	. . . . . . 7 ⊢ (rank‘𝑥) ∈ V
38	37	elsuc 6418	. . . . . 6 ⊢ ((rank‘𝑥) ∈ suc 𝐴 ↔ ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
39	36, 38	sylib 220	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
40	39	orcomd 882	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴))
41		fvex 6880	. . . . . . . 8 ⊢ (𝑅1‘𝐴) ∈ V
42		elpwi 4562	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → 𝑥 ⊆ (𝑅1‘𝐴))
43	42	ad2antlr 737	. . . . . . . 8 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ⊆ (𝑅1‘𝐴))
44		ssdomg 8981	. . . . . . . 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 10735	. . . . . . . . . 10 ⊢ ¬ 𝑥 ≺ (cf‘(rank‘𝑥))
47		fveq2 6867	. . . . . . . . . . . 12 ⊢ ((rank‘𝑥) = 𝐴 → (cf‘(rank‘𝑥)) = (cf‘𝐴))
48		elina 10645	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
49	48	simp2bi 1159	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
50	47, 49	sylan9eqr 2819	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (cf‘(rank‘𝑥)) = 𝐴)
51	50	breq2d 5112	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (cf‘(rank‘𝑥)) ↔ 𝑥 ≺ 𝐴))
52	46, 51	mtbii 328	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ 𝐴)
53		inar1 10733	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ≈ 𝐴)
54		sdomentr 9083	. . . . . . . . . . . 12 ⊢ ((𝑥 ≺ (𝑅1‘𝐴) ∧ (𝑅1‘𝐴) ≈ 𝐴) → 𝑥 ≺ 𝐴)
55	54	expcom 417	. . . . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (𝑅1‘𝐴) → 𝑥 ≺ 𝐴))
58	52, 57	mtod 200	. . . . . . . 8 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1‘𝐴))
59	58	adantlr 725	. . . . . . 7 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1‘𝐴))
60		bren2 8964	. . . . . . 7 ⊢ (𝑥 ≈ (𝑅1‘𝐴) ↔ (𝑥 ≼ (𝑅1‘𝐴) ∧ ¬ 𝑥 ≺ (𝑅1‘𝐴)))
61	45, 59, 60	sylanbrc 592	. . . . . 6 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≈ (𝑅1‘𝐴))
62	61	ex 416	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) = 𝐴 → 𝑥 ≈ (𝑅1‘𝐴)))
63		r1elwf 9754	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑅1‘suc 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
64	33, 63	biimtrrdi 256	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On)))
65	64	imp 410	. . . . . . 7 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
66		r1fnon 9725	. . . . . . . . . 10 ⊢ 𝑅1 Fn On
67	66	fndmi 6625	. . . . . . . . 9 ⊢ dom 𝑅1 = On
68	30, 67	eleqtrrdi 2873	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ dom 𝑅1)
69	68	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → 𝐴 ∈ dom 𝑅1)
70		rankr1ag 9760	. . . . . . 7 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝐴 ∈ dom 𝑅1) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
71	65, 69, 70	syl2anc 593	. . . . . 6 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (𝑥 ∈ (𝑅1‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
72	71	biimprd 250	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) ∈ 𝐴 → 𝑥 ∈ (𝑅1‘𝐴)))
73	62, 72	orim12d 977	. . . 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 3154	. 2 ⊢ (𝐴 ∈ Inacc → ∀𝑥 ∈ 𝒫 (𝑅1‘𝐴)(𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴)))
76		eltsk2g 10709	. . 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 592	1 ⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ∈ Tarski)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∨ wo 858   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  ∀wral 3076  ∃wrex 3086  Vcvv 3454   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4555  ∪ cuni 4865  ∪ ciun 4949   class class class wbr 5100  dom cdm 5647   “ cima 5650  Oncon0 6346  Lim wlim 6347  suc csuc 6348  ‘cfv 6521   ≈ cen 8924   ≼ cdom 8925   ≺ csdm 8926  𝑅₁cr1 9720  rankcrnk 9721  cfccf 9895  Inacc_wcwina 10640  Inacccina 10641  Tarskictsk 10706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pow 5322  ax-pr 5390  ax-un 7718  ax-inf2 9596  ax-ac2 10420

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-iin 4952  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-isom 6530  df-riota 7353  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-1st 7970  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8678  df-map 8810  df-en 8928  df-dom 8929  df-sdom 8930  df-fin 8931  df-oi 9458  df-r1 9722  df-rank 9723  df-card 9897  df-cf 9899  df-acn 9900  df-ac 10072  df-wina 10642  df-ina 10643  df-tsk 10707

This theorem is referenced by:  r1omtsk  10737  r1tskina  10740  grutsk  10780  inagrud  44872