Theorem onnseq 7975

Description: There are no length ω decreasing sequences in the ordinals. See also noinfep 9117 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 7491	. . . . 5 ⊢ E We On
2		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
3	2	eleq1d 2897	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘∅) ∈ On))
4		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
5	4	eleq1d 2897	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘𝑧) ∈ On))
6		fveq2 6664	. . . . . . . . . . 11 ⊢ (𝑦 = suc 𝑧 → (𝐹‘𝑦) = (𝐹‘suc 𝑧))
7	6	eleq1d 2897	. . . . . . . . . 10 ⊢ (𝑦 = suc 𝑧 → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘suc 𝑧) ∈ On))
8		simpl 485	. . . . . . . . . 10 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝐹‘∅) ∈ On)
9		suceq 6250	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
10	9	fveq2d 6668	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑧))
11		fveq2 6664	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
12	10, 11	eleq12d 2907	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
13	12	rspcv 3617	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
14		onelon 6210	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑧) ∈ On ∧ (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)) → (𝐹‘suc 𝑧) ∈ On)
15	14	expcom 416	. . . . . . . . . . . 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 493	. . . . . . . . . 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 7604	. . . . . . . . 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 3181	. . . . . . 7 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∀𝑦 ∈ ω (𝐹‘𝑦) ∈ On)
21		eqid 2821	. . . . . . . 8 ⊢ (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = (𝑦 ∈ ω ↦ (𝐹‘𝑦))
22	21	fmpt 6868	. . . . . . 7 ⊢ (∀𝑦 ∈ ω (𝐹‘𝑦) ∈ On ↔ (𝑦 ∈ ω ↦ (𝐹‘𝑦)):ω⟶On)
23	20, 22	sylib 220	. . . . . 6 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝑦 ∈ ω ↦ (𝐹‘𝑦)):ω⟶On)
24	23	frnd 6515	. . . . 5 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ⊆ On)
25		peano1 7595	. . . . . . . 8 ⊢ ∅ ∈ ω
26	23	fdmd 6517	. . . . . . . 8 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ω)
27	25, 26	eleqtrrid 2920	. . . . . . 7 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∅ ∈ dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)))
28	27	ne0d 4300	. . . . . 6 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅)
29		dm0rn0 5789	. . . . . . 7 ⊢ (dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ∅)
30	29	necon3bii 3068	. . . . . 6 ⊢ (dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅)
31	28, 30	sylib 220	. . . . 5 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅)
32		wefrc 5543	. . . . 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 1462	. . . 4 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦))(ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅)
34		fvex 6677	. . . . . 6 ⊢ (𝐹‘𝑤) ∈ V
35	34	rgenw 3150	. . . . 5 ⊢ ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ V
36		fveq2 6664	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
37	36	cbvmptv 5161	. . . . . 6 ⊢ (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = (𝑤 ∈ ω ↦ (𝐹‘𝑤))
38		ineq2 4182	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑤) → (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)))
39	38	eqeq1d 2823	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑤) → ((ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅ ↔ (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅))
40	37, 39	rexrnmptw 6855	. . . . 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 220	. . 3 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)
43		peano2 7596	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → suc 𝑤 ∈ ω)
44	43	adantl 484	. . . . . . . 8 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → suc 𝑤 ∈ ω)
45		eqid 2821	. . . . . . . 8 ⊢ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)
46		fveq2 6664	. . . . . . . . 9 ⊢ (𝑦 = suc 𝑤 → (𝐹‘𝑦) = (𝐹‘suc 𝑤))
47	46	rspceeqv 3637	. . . . . . . 8 ⊢ ((suc 𝑤 ∈ ω ∧ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦))
48	44, 45, 47	sylancl 588	. . . . . . 7 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦))
49		fvex 6677	. . . . . . . 8 ⊢ (𝐹‘suc 𝑤) ∈ V
50	21	elrnmpt 5822	. . . . . . . 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 236	. . . . . 6 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)))
53		suceq 6250	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤)
54	53	fveq2d 6668	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑤))
55		fveq2 6664	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
56	54, 55	eleq12d 2907	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤)))
57	56	rspccva 3621	. . . . . . 7 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤))
58	57	adantll 712	. . . . . 6 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤))
59		inelcm 4413	. . . . . 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 586	. . . . 5 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) ≠ ∅)
61	60	neneqd 3021	. . . 4 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → ¬ (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)
62	61	nrexdv 3270	. . 3 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ¬ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)
63	42, 62	pm2.65da 815	. 2 ⊢ ((𝐹‘∅) ∈ On → ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
64		rexnal 3238	. 2 ⊢ (∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
65	63, 64	sylibr 236	1 ⊢ ((𝐹‘∅) ∈ On → ∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  Vcvv 3494   ∩ cin 3934   ⊆ wss 3935  ∅c0 4290   ↦ cmpt 5138   E cep 5458   We wwe 5507  dom cdm 5549  ran crn 5550  Oncon0 6185  suc csuc 6187  ⟶wf 6345  ‘cfv 6349  ωcom 7574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-om 7575

This theorem is referenced by: (None)