Theorem onnseq 8278

Description: There are no length ω decreasing sequences in the ordinals. See also noinfep 9573 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 7722	. . . . 5 ⊢ E We On
2		fveq2 6835	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
3	2	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘∅) ∈ On))
4		fveq2 6835	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
5	4	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘𝑧) ∈ On))
6		fveq2 6835	. . . . . . . . . . 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 482	. . . . . . . . . 10 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝐹‘∅) ∈ On)
9		suceq 6386	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
10	9	fveq2d 6839	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑧))
11		fveq2 6835	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧))
12	10, 11	eleq12d 2831	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → ((𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
13	12	rspcv 3573	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
14		onelon 6343	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑧) ∈ On ∧ (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)) → (𝐹‘suc 𝑧) ∈ On)
15	14	expcom 413	. . . . . . . . . . . 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 490	. . . . . . . . . 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 7842	. . . . . . . . 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 3128	. . . . . . 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 7057	. . . . . . 7 ⊢ (∀𝑦 ∈ ω (𝐹‘𝑦) ∈ On ↔ (𝑦 ∈ ω ↦ (𝐹‘𝑦)):ω⟶On)
23	20, 22	sylib 218	. . . . . 6 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝑦 ∈ ω ↦ (𝐹‘𝑦)):ω⟶On)
24	23	frnd 6671	. . . . 5 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ⊆ On)
25		peano1 7833	. . . . . . . 8 ⊢ ∅ ∈ ω
26	23	fdmd 6673	. . . . . . . 8 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ω)
27	25, 26	eleqtrrid 2844	. . . . . . 7 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∅ ∈ dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)))
28	27	ne0d 4295	. . . . . 6 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅)
29		dm0rn0 5874	. . . . . . 7 ⊢ (dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ∅)
30	29	necon3bii 2985	. . . . . 6 ⊢ (dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅)
31	28, 30	sylib 218	. . . . 5 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅)
32		wefrc 5619	. . . . 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 1469	. . . 4 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦))(ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅)
34		fvex 6848	. . . . . 6 ⊢ (𝐹‘𝑤) ∈ V
35	34	rgenw 3056	. . . . 5 ⊢ ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ V
36		fveq2 6835	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
37	36	cbvmptv 5203	. . . . . 6 ⊢ (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = (𝑤 ∈ ω ↦ (𝐹‘𝑤))
38		ineq2 4167	. . . . . . 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 7042	. . . . 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 218	. . 3 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)
43		peano2 7834	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → suc 𝑤 ∈ ω)
44	43	adantl 481	. . . . . . . 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 6835	. . . . . . . . 9 ⊢ (𝑦 = suc 𝑤 → (𝐹‘𝑦) = (𝐹‘suc 𝑤))
47	46	rspceeqv 3600	. . . . . . . 8 ⊢ ((suc 𝑤 ∈ ω ∧ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦))
48	44, 45, 47	sylancl 587	. . . . . . 7 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦))
49		fvex 6848	. . . . . . . 8 ⊢ (𝐹‘suc 𝑤) ∈ V
50	21	elrnmpt 5908	. . . . . . . 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 234	. . . . . 6 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)))
53		suceq 6386	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤)
54	53	fveq2d 6839	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑤))
55		fveq2 6835	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤))
56	54, 55	eleq12d 2831	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ((𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤)))
57	56	rspccva 3576	. . . . . . 7 ⊢ ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤))
58	57	adantll 715	. . . . . 6 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹‘𝑤))
59		inelcm 4418	. . . . . 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 585	. . . . 5 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) ≠ ∅)
61	60	neneqd 2938	. . . 4 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → ¬ (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)
62	61	nrexdv 3132	. . 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 3089	. 2 ⊢ (∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
65	63, 64	sylibr 234	1 ⊢ ((𝐹‘∅) ∈ On → ∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3061  Vcvv 3441   ∩ cin 3901   ⊆ wss 3902  ∅c0 4286   ↦ cmpt 5180   E cep 5524   We wwe 5577  dom cdm 5625  ran crn 5626  Oncon0 6318  suc csuc 6320  ⟶wf 6489  ‘cfv 6493  ωcom 7810

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-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

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 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  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-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fv 6501  df-om 7811

This theorem is referenced by: (None)