Theorem inatsk 10695

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 10607	. . . . . 6 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ Inaccw)
2		winaon 10605	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inaccw → 𝐴 ∈ On)
3		winalim 10612	. . . . . . . . . 10 ⊢ (𝐴 ∈ Inaccw → Lim 𝐴)
4		r1lim 9690	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ Lim 𝐴) → (𝑅1‘𝐴) = ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦))
5	2, 3, 4	syl2anc 585	. . . . . . . . 9 ⊢ (𝐴 ∈ Inaccw → (𝑅1‘𝐴) = ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦))
6	5	eleq2d 2823	. . . . . . . 8 ⊢ (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1‘𝐴) ↔ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦)))
7		eliun 4938	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 (𝑅1‘𝑦) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦))
8	6, 7	bitrdi 287	. . . . . . 7 ⊢ (𝐴 ∈ Inaccw → (𝑥 ∈ (𝑅1‘𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦)))
9		onelon 6343	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
10	2, 9	sylan 581	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ On)
11		r1pw 9763	. . . . . . . . . 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 7794	. . . . . . . . . . . . 13 ⊢ (Lim 𝐴 → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
14	3, 13	syl 17	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Inaccw → (𝑦 ∈ 𝐴 ↔ suc 𝑦 ∈ 𝐴))
15		r1ord2 9699	. . . . . . . . . . . . 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 240	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inaccw → (𝑦 ∈ 𝐴 → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴)))
18	17	imp 406	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝑅1‘suc 𝑦) ⊆ (𝑅1‘𝐴))
19	18	sseld 3921	. . . . . . . . 9 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝒫 𝑥 ∈ (𝑅1‘suc 𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
20	12, 19	sylbid 240	. . . . . . . 8 ⊢ ((𝐴 ∈ Inaccw ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ (𝑅1‘𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
21	20	rexlimdva 3139	. . . . . . 7 ⊢ (𝐴 ∈ Inaccw → (∃𝑦 ∈ 𝐴 𝑥 ∈ (𝑅1‘𝑦) → 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
22	8, 21	sylbid 240	. . . . . 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 406	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1‘𝐴)) → 𝒫 𝑥 ∈ (𝑅1‘𝐴))
25		elssuni 4882	. . . . 5 ⊢ (𝒫 𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ⊆ ∪ (𝑅1‘𝐴))
26		r1tr2 9695	. . . . 5 ⊢ ∪ (𝑅1‘𝐴) ⊆ (𝑅1‘𝐴)
27	25, 26	sstrdi 3935	. . . 4 ⊢ (𝒫 𝑥 ∈ (𝑅1‘𝐴) → 𝒫 𝑥 ⊆ (𝑅1‘𝐴))
28	24, 27	jccil 522	. . 3 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ (𝑅1‘𝐴)) → (𝒫 𝑥 ⊆ (𝑅1‘𝐴) ∧ 𝒫 𝑥 ∈ (𝑅1‘𝐴)))
29	28	ralrimiva 3130	. 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 9688	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → (𝑅1‘suc 𝐴) = 𝒫 (𝑅1‘𝐴))
32	31	eleq2d 2823	. . . . . . . . 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 9716	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑅1‘suc 𝐴) → (rank‘𝑥) ∈ suc 𝐴)
35	33, 34	biimtrrdi 254	. . . . . . 7 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → (rank‘𝑥) ∈ suc 𝐴))
36	35	imp 406	. . . . . 6 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → (rank‘𝑥) ∈ suc 𝐴)
37		fvex 6848	. . . . . . 7 ⊢ (rank‘𝑥) ∈ V
38	37	elsuc 6390	. . . . . 6 ⊢ ((rank‘𝑥) ∈ suc 𝐴 ↔ ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
39	36, 38	sylib 218	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) ∈ 𝐴 ∨ (rank‘𝑥) = 𝐴))
40	39	orcomd 872	. . . 4 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) = 𝐴 ∨ (rank‘𝑥) ∈ 𝐴))
41		fvex 6848	. . . . . . . 8 ⊢ (𝑅1‘𝐴) ∈ V
42		elpwi 4549	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → 𝑥 ⊆ (𝑅1‘𝐴))
43	42	ad2antlr 728	. . . . . . . 8 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → 𝑥 ⊆ (𝑅1‘𝐴))
44		ssdomg 8941	. . . . . . . 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 10694	. . . . . . . . . 10 ⊢ ¬ 𝑥 ≺ (cf‘(rank‘𝑥))
47		fveq2 6835	. . . . . . . . . . . 12 ⊢ ((rank‘𝑥) = 𝐴 → (cf‘(rank‘𝑥)) = (cf‘𝐴))
48		elina 10604	. . . . . . . . . . . . 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 2794	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → (cf‘(rank‘𝑥)) = 𝐴)
51	50	breq2d 5098	. . . . . . . . . 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 10692	. . . . . . . . . . 11 ⊢ (𝐴 ∈ Inacc → (𝑅1‘𝐴) ≈ 𝐴)
54		sdomentr 9043	. . . . . . . . . . . 12 ⊢ ((𝑥 ≺ (𝑅1‘𝐴) ∧ (𝑅1‘𝐴) ≈ 𝐴) → 𝑥 ≺ 𝐴)
55	54	expcom 413	. . . . . . . . . . 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 198	. . . . . . . 8 ⊢ ((𝐴 ∈ Inacc ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1‘𝐴))
59	58	adantlr 716	. . . . . . 7 ⊢ (((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) ∧ (rank‘𝑥) = 𝐴) → ¬ 𝑥 ≺ (𝑅1‘𝐴))
60		bren2 8924	. . . . . . 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 412	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) = 𝐴 → 𝑥 ≈ (𝑅1‘𝐴)))
63		r1elwf 9714	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑅1‘suc 𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On))
64	33, 63	biimtrrdi 254	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → (𝑥 ∈ 𝒫 (𝑅1‘𝐴) → 𝑥 ∈ ∪ (𝑅1 “ On)))
65	64	imp 406	. . . . . . 7 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → 𝑥 ∈ ∪ (𝑅1 “ On))
66		r1fnon 9685	. . . . . . . . . 10 ⊢ 𝑅1 Fn On
67	66	fndmi 6597	. . . . . . . . 9 ⊢ dom 𝑅1 = On
68	30, 67	eleqtrrdi 2848	. . . . . . . 8 ⊢ (𝐴 ∈ Inacc → 𝐴 ∈ dom 𝑅1)
69	68	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → 𝐴 ∈ dom 𝑅1)
70		rankr1ag 9720	. . . . . . 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 248	. . . . 5 ⊢ ((𝐴 ∈ Inacc ∧ 𝑥 ∈ 𝒫 (𝑅1‘𝐴)) → ((rank‘𝑥) ∈ 𝐴 → 𝑥 ∈ (𝑅1‘𝐴)))
73	62, 72	orim12d 967	. . . 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 3130	. 2 ⊢ (𝐴 ∈ Inacc → ∀𝑥 ∈ 𝒫 (𝑅1‘𝐴)(𝑥 ≈ (𝑅1‘𝐴) ∨ 𝑥 ∈ (𝑅1‘𝐴)))
76		eltsk2g 10668	. . 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)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  ∪ cuni 4851  ∪ ciun 4934   class class class wbr 5086  dom cdm 5625   “ cima 5628  Oncon0 6318  Lim wlim 6319  suc csuc 6320  ‘cfv 6493   ≈ cen 8884   ≼ cdom 8885   ≺ csdm 8886  𝑅₁cr1 9680  rankcrnk 9681  cfccf 9855  Inacc_wcwina 10599  Inacccina 10600  Tarskictsk 10665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-inf2 9556  ax-ac2 10379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-isom 6502  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-2o 8400  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-oi 9419  df-r1 9682  df-rank 9683  df-card 9857  df-cf 9859  df-acn 9860  df-ac 10032  df-wina 10601  df-ina 10602  df-tsk 10666

This theorem is referenced by:  r1omtsk  10696  r1tskina  10699  grutsk  10739  inagrud  44744