Theorem onnseq 8081

Description: There are no length ω decreasing sequences in the ordinals. See also noinfep 9275 for a stronger version assuming Regularity. (Contributed by Mario Carneiro, 19-May-2015.)

Assertion

Ref	Expression
onnseq	⊢ ((𝐹‘∅) ∈ On → ∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))

Distinct variable group:   𝑥,𝐹

Proof of Theorem onnseq

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		epweon 7560	. . . . 5 ⊢ E We On
2		fveq2 6717	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
3	2	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘∅) ∈ On))
4		fveq2 6717	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
5	4	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘𝑧) ∈ On))
6		fveq2 6717	. . . . . . . . . . 11 ⊢ (𝑦 = suc 𝑧 → (𝐹‘𝑦) = (𝐹‘suc 𝑧))
7	6	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑦 = suc 𝑧 → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘suc 𝑧) ∈ On))
8		simpl 486	. . . . . . . . . 10 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝐹‘∅) ∈ On)
9		suceq 6278	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
10	9	fveq2d 6721	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑧))
11		fveq2 6717	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
12	10, 11	eleq12d 2832	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
13	12	rspcv 3532	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
14		onelon 6238	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑧) ∈ On ∧ (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)) → (𝐹‘suc 𝑧) ∈ On)
15	14	expcom 417	. . . . . . . . . . . 12 ⊢ ((𝐹‘suc 𝑧) ∈ (𝐹‘𝑧) → ((𝐹‘𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On))
16	13, 15	syl6 35	. . . . . . . . . . 11 ⊢ (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → ((𝐹‘𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On)))
17	16	adantld 494	. . . . . . . . . 10 ⊢ (𝑧 ∈ ω → (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ((𝐹‘𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On)))
18	3, 5, 7, 8, 17	finds2 7678	. . . . . . . . 9 ⊢ (𝑦 ∈ ω → (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝐹‘𝑦) ∈ On))
19	18	com12 32	. . . . . . . 8 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝑦 ∈ ω → (𝐹‘𝑦) ∈ On))
20	19	ralrimiv 3104	. . . . . . 7 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∀𝑦 ∈ ω (𝐹‘𝑦) ∈ On)
21		eqid 2737	. . . . . . . 8 ⊢ (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = (𝑦 ∈ ω ↦ (𝐹‘𝑦))
22	21	fmpt 6927	. . . . . . 7 ⊢ (∀𝑦 ∈ ω (𝐹‘𝑦) ∈ On ↔ (𝑦 ∈ ω ↦ (𝐹‘𝑦)):ω⟶On)
23	20, 22	sylib 221	. . . . . 6 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝑦 ∈ ω ↦ (𝐹‘𝑦)):ω⟶On)
24	23	frnd 6553	. . . . 5 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ⊆ On)
25		peano1 7667	. . . . . . . 8 ⊢ ∅ ∈ ω
26	23	fdmd 6556	. . . . . . . 8 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ω)
27	25, 26	eleqtrrid 2845	. . . . . . 7 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∅ ∈ dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)))
28	27	ne0d 4250	. . . . . 6 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅)
29		dm0rn0 5794	. . . . . . 7 ⊢ (dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ∅)
30	29	necon3bii 2993	. . . . . 6 ⊢ (dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅)
31	28, 30	sylib 221	. . . . 5 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅)
32		wefrc 5545	. . . . 5 ⊢ (( E We On ∧ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ⊆ On ∧ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅) → ∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦))(ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅)
33	1, 24, 31, 32	mp3an2i 1468	. . . 4 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦))(ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅)
34		fvex 6730	. . . . . 6 ⊢ (𝐹‘𝑤) ∈ V
35	34	rgenw 3073	. . . . 5 ⊢ ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ V
36		fveq2 6717	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
37	36	cbvmptv 5158	. . . . . 6 ⊢ (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = (𝑤 ∈ ω ↦ (𝐹‘𝑤))
38		ineq2 4121	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑤) → (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)))
39	38	eqeq1d 2739	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑤) → ((ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅ ↔ (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅))
40	37, 39	rexrnmptw 6914	. . . . 5 ⊢ (∀𝑤 ∈ ω (𝐹‘𝑤) ∈ V → (∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦))(ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅ ↔ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅))
41	35, 40	ax-mp 5	. . . 4 ⊢ (∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦))(ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅ ↔ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)
42	33, 41	sylib 221	. . 3 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)
43		peano2 7668	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → suc 𝑤 ∈ ω)
44	43	adantl 485	. . . . . . . 8 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → suc 𝑤 ∈ ω)
45		eqid 2737	. . . . . . . 8 ⊢ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)
46		fveq2 6717	. . . . . . . . 9 ⊢ (𝑦 = suc 𝑤 → (𝐹‘𝑦) = (𝐹‘suc 𝑤))
47	46	rspceeqv 3552	. . . . . . . 8 ⊢ ((suc 𝑤 ∈ ω ∧ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦))
48	44, 45, 47	sylancl 589	. . . . . . 7 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦))
49		fvex 6730	. . . . . . . 8 ⊢ (𝐹‘suc 𝑤) ∈ V
50	21	elrnmpt 5825	. . . . . . . 8 ⊢ ((𝐹‘suc 𝑤) ∈ V → ((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ↔ ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦)))
51	49, 50	ax-mp 5	. . . . . . 7 ⊢ ((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ↔ ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦))
52	48, 51	sylibr 237	. . . . . 6 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)))
53		suceq 6278	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤)
54	53	fveq2d 6721	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑤))
55		fveq2 6717	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
56	54, 55	eleq12d 2832	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤)))
57	56	rspccva 3536	. . . . . . 7 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤))
58	57	adantll 714	. . . . . 6 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤))
59		inelcm 4379	. . . . . 6 ⊢ (((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∧ (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤)) → (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) ≠ ∅)
60	52, 58, 59	syl2anc 587	. . . . 5 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) ≠ ∅)
61	60	neneqd 2945	. . . 4 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → ¬ (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)
62	61	nrexdv 3189	. . 3 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ¬ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)
63	42, 62	pm2.65da 817	. 2 ⊢ ((𝐹‘∅) ∈ On → ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
64		rexnal 3160	. 2 ⊢ (∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
65	63, 64	sylibr 237	1 ⊢ ((𝐹‘∅) ∈ On → ∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110   ≠ wne 2940  ∀wral 3061  ∃wrex 3062  Vcvv 3408   ∩ cin 3865   ⊆ wss 3866  ∅c0 4237   ↦ cmpt 5135   E cep 5459   We wwe 5508  dom cdm 5551  ran crn 5552  Oncon0 6213  suc csuc 6215  ⟶wf 6376  ‘cfv 6380  ωcom 7644

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322  ax-un 7523

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388  df-om 7645

This theorem is referenced by: (None)