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Theorem inatsk 10770

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 10682	. . . . . 6 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inacc_w)
2		winaon 10680	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inacc_w → 𝐴 ∈ On)
3		winalim 10687	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inacc_w → Lim 𝐴)
4		r1lim 9764	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑅₁‘𝐴) = ∪ 𝑦 ∈ 𝐴 (𝑅₁‘𝑦))
5	2, 3, 4	syl2anc 585	. . . . . . . . 9 ⊢ (𝐴 ∈ Inacc_w → (𝑅₁‘𝐴) = ∪ 𝑦 ∈ 𝐴 (𝑅₁‘𝑦))
6	5	eleq2d 2820	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc_w → (𝑥 ∈ (𝑅₁‘𝐴) ↔ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (𝑅₁‘𝑦)))
7		eliun 5001	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (𝑅₁‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅₁‘𝑦))
8	6, 7	bitrdi 287	. . . . . . 7 ⊢ (𝐴 ∈ Inacc_w → (𝑥 ∈ (𝑅₁‘𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅₁‘𝑦)))
9		onelon 6387	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
10	2, 9	sylan 581	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inacc_w ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
11		r1pw 9837	. . . . . . . . . 10 ⊢ (𝑦 ∈ On → (𝑥 ∈ (𝑅₁‘𝑦) ↔ 𝒫 𝑥 ∈ (𝑅₁‘suc 𝑦)))
12	10, 11	syl 17	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inacc_w ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (𝑅₁‘𝑦) ↔ 𝒫 𝑥 ∈ (𝑅₁‘suc 𝑦)))
13		limsuc 7835	. . . . . . . . . . . . 13 ⊢ (Lim 𝐴 → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
14	3, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inacc_w → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
15		r1ord2 9773	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ On → (suc 𝑦 ∈ 𝐴 → (𝑅₁‘suc 𝑦) ⊆ (𝑅₁‘𝐴)))
16	2, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inacc_w → (suc 𝑦 ∈ 𝐴 → (𝑅₁‘suc 𝑦) ⊆ (𝑅₁‘𝐴)))
17	14, 16	sylbid 239	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inacc_w → (𝑦 ∈ 𝐴 → (𝑅₁‘suc 𝑦) ⊆ (𝑅₁‘𝐴)))
18	17	imp 408	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inacc_w ∧ 𝑦 ∈ 𝐴) → (𝑅₁‘suc 𝑦) ⊆ (𝑅₁‘𝐴))
19	18	sseld 3981	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inacc_w ∧ 𝑦 ∈ 𝐴) → (𝒫 𝑥 ∈ (𝑅₁‘suc 𝑦) → 𝒫 𝑥 ∈ (𝑅₁‘𝐴)))
20	12, 19	sylbid 239	. . . . . . . 8 ⊢ ((𝐴 ∈ Inacc_w ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (𝑅₁‘𝑦) → 𝒫 𝑥 ∈ (𝑅₁‘𝐴)))
21	20	rexlimdva 3156	. . . . . . 7 ⊢ (𝐴 ∈ Inacc_w → (∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅₁‘𝑦) → 𝒫 𝑥 ∈ (𝑅₁‘𝐴)))
22	8, 21	sylbid 239	. . . . . 6 ⊢ (𝐴 ∈ Inacc_w → (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 408	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → 𝒫 𝑥 ∈ (𝑅₁‘𝐴))
25		elssuni 4941	. . . . 5 ⊢ (𝒫 𝑥 ∈ (𝑅₁‘𝐴) → 𝒫 𝑥 ⊆ ∪ (𝑅₁‘𝐴))
26		r1tr2 9769	. . . . 5 ⊢ ∪ (𝑅₁‘𝐴) ⊆ (𝑅₁‘𝐴)
27	25, 26	sstrdi 3994	. . . 4 ⊢ (𝒫 𝑥 ∈ (𝑅₁‘𝐴) → 𝒫 𝑥 ⊆ (𝑅₁‘𝐴))
28	24, 27	jccil 524	. . 3 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅₁‘𝐴)) → (𝒫 𝑥 ⊆ (𝑅₁‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅₁‘𝐴)))
29	28	ralrimiva 3147	. 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 9762	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑅₁‘suc 𝐴) = 𝒫 (𝑅₁‘𝐴))
32	31	eleq2d 2820	. . . . . . . . 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 9790	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅₁‘suc 𝐴) → (rank‘𝑥) ∈ suc 𝐴)
35	33, 34	syl6bir 254	. . . . . . 7 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅₁‘𝐴) → (rank‘𝑥) ∈ suc 𝐴))
36	35	imp 408	. . . . . 6 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅₁‘𝐴)) → (rank‘𝑥) ∈ suc 𝐴)
37		fvex 6902	. . . . . . 7 ⊢ (rank‘𝑥) ∈ V
38	37	elsuc 6432	. . . . . 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 870	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅₁‘𝐴)) → ((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴))
41		fvex 6902	. . . . . . . 8 ⊢ (𝑅₁‘𝐴) ∈ V
42		elpwi 4609	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑅₁‘𝐴) → 𝑥 ⊆ (𝑅₁‘𝐴))
43	42	ad2antlr 726	. . . . . . . 8 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅₁‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ⊆ (𝑅₁‘𝐴))
44		ssdomg 8993	. . . . . . . 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 10769	. . . . . . . . . 10 ⊢ ¬ 𝑥 ≺ (cf‘(rank‘𝑥))
47		fveq2 6889	. . . . . . . . . . . 12 ⊢ ((rank‘𝑥) = 𝐴 → (cf‘(rank‘𝑥)) = (cf‘𝐴))
48		elina 10679	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Inacc ↔ (𝐴 ≠ ∅ ∧ (cf‘𝐴) = 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝒫 𝑥 ≺ 𝐴))
49	48	simp2bi 1147	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inacc → (cf‘𝐴) = 𝐴)
50	47, 49	sylan9eqr 2795	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (cf‘(rank‘𝑥)) = 𝐴)
51	50	breq2d 5160	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (𝑥 ≺ (cf‘(rank‘𝑥)) ↔ 𝑥 ≺ 𝐴))
52	46, 51	mtbii 326	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ 𝐴)
53		inar1 10767	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inacc → (𝑅₁‘𝐴) ≈ 𝐴)
54		sdomentr 9108	. . . . . . . . . . . 12 ⊢ ((𝑥 ≺ (𝑅₁‘𝐴) ∧ (𝑅₁‘𝐴) ≈ 𝐴) → 𝑥 ≺ 𝐴)
55	54	expcom 415	. . . . . . . . . . 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 482	. . . . . . . . 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 714	. . . . . . 7 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅₁‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅₁‘𝐴))
60		bren2 8976	. . . . . . 7 ⊢ (𝑥 ≈ (𝑅₁‘𝐴) ↔ (𝑥 ≼ (𝑅₁‘𝐴) ∧ ¬ 𝑥 ≺ (𝑅₁‘𝐴)))
61	45, 59, 60	sylanbrc 584	. . . . . 6 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅₁‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ≈ (𝑅₁‘𝐴))
62	61	ex 414	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅₁‘𝐴)) → ((rank‘𝑥) = 𝐴 → 𝑥 ≈ (𝑅₁‘𝐴)))
63		r1elwf 9788	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑅₁‘suc 𝐴) → 𝑥 ∈ ∪ (𝑅₁ “ On))
64	33, 63	syl6bir 254	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅₁‘𝐴) → 𝑥 ∈ ∪ (𝑅₁ “ On)))
65	64	imp 408	. . . . . . 7 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅₁‘𝐴)) → 𝑥 ∈ ∪ (𝑅₁ “ On))
66		r1fnon 9759	. . . . . . . . . 10 ⊢ 𝑅₁ Fn On
67	66	fndmi 6651	. . . . . . . . 9 ⊢ dom 𝑅₁ = On
68	30, 67	eleqtrrdi 2845	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ dom 𝑅₁)
69	68	adantr 482	. . . . . . 7 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅₁‘𝐴)) → 𝐴 ∈ dom 𝑅₁)
70		rankr1ag 9794	. . . . . . 7 ⊢ ((𝑥 ∈ ∪ (𝑅₁ “ On) ∧ 𝐴 ∈ dom 𝑅₁) → (𝑥 ∈ (𝑅₁‘𝐴) ↔ (rank‘𝑥) ∈ 𝐴))
71	65, 69, 70	syl2anc 585	. . . . . 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 964	. . . 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 3147	. 2 ⊢ (𝐴 ∈ Inacc → ∀𝑥 ∈ 𝒫 (𝑅₁‘𝐴)(𝑥 ≈ (𝑅₁‘𝐴) ∨ 𝑥 ∈ (𝑅₁‘𝐴)))
76		eltsk2g 10743	. . 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 584	1 ⊢ (𝐴 ∈ Inacc → (𝑅₁‘𝐴) ∈ Tarski)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∨ wo 846 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 ∪ cuni 4908 ∪ ciun 4997 class class class wbr 5148 dom cdm 5676 “ cima 5679 Oncon0 6362 Lim wlim 6363 suc csuc 6364 ‘cfv 6541 ≈ cen 8933 ≼ cdom 8934 ≺ csdm 8935 𝑅₁cr1 9754 rankcrnk 9755 cfccf 9929 Inacc_wcwina 10674 Inacccina 10675 Tarskictsk 10740

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-inf2 9633 ax-ac2 10455

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-2o 8464 df-er 8700 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-oi 9502 df-r1 9756 df-rank 9757 df-card 9931 df-cf 9933 df-acn 9934 df-ac 10108 df-wina 10676 df-ina 10677 df-tsk 10741

This theorem is referenced by: r1omtsk 10771 r1tskina 10774 grutsk 10814 inagrud 43041

W3C validator