Theorem onnseq 8290

Description: There are no length ω decreasing sequences in the ordinals. See also noinfep 9596 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 7709	. . . . 5 ⊢ E We On
2		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
3	2	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘∅) ∈ On))
4		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
5	4	eleq1d 2822	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) ∈ On ↔ (𝐹‘𝑧) ∈ On))
6		fveq2 6842	. . . . . . . . . . 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 483	. . . . . . . . . 10 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝐹‘∅) ∈ On)
9		suceq 6383	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
10	9	fveq2d 6846	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑧 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑧))
11		fveq2 6842	. . . . . . . . . . . . . 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 3577	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) → (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)))
14		onelon 6342	. . . . . . . . . . . . 13 ⊢ (((𝐹‘𝑧) ∈ On ∧ (𝐹‘suc 𝑧) ∈ (𝐹‘𝑧)) → (𝐹‘suc 𝑧) ∈ On)
15	14	expcom 414	. . . . . . . . . . . 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 491	. . . . . . . . . 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 7837	. . . . . . . . 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 3142	. . . . . . 7 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∀𝑦 ∈ ω (𝐹‘𝑦) ∈ On)
21		eqid 2736	. . . . . . . 8 ⊢ (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = (𝑦 ∈ ω ↦ (𝐹‘𝑦))
22	21	fmpt 7058	. . . . . . 7 ⊢ (∀𝑦 ∈ ω (𝐹‘𝑦) ∈ On ↔ (𝑦 ∈ ω ↦ (𝐹‘𝑦)):ω⟶On)
23	20, 22	sylib 217	. . . . . 6 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → (𝑦 ∈ ω ↦ (𝐹‘𝑦)):ω⟶On)
24	23	frnd 6676	. . . . 5 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ⊆ On)
25		peano1 7825	. . . . . . . 8 ⊢ ∅ ∈ ω
26	23	fdmd 6679	. . . . . . . 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 4295	. . . . . 6 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅)
29		dm0rn0 5880	. . . . . . 7 ⊢ (dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = ∅)
30	29	necon3bii 2996	. . . . . 6 ⊢ (dom (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅)
31	28, 30	sylib 217	. . . . 5 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ≠ ∅)
32		wefrc 5627	. . . . 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 1466	. . . 4 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦))(ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅)
34		fvex 6855	. . . . . 6 ⊢ (𝐹‘𝑤) ∈ V
35	34	rgenw 3068	. . . . 5 ⊢ ∀𝑤 ∈ ω (𝐹‘𝑤) ∈ V
36		fveq2 6842	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝐹‘𝑦) = (𝐹‘𝑤))
37	36	cbvmptv 5218	. . . . . 6 ⊢ (𝑦 ∈ ω ↦ (𝐹‘𝑦)) = (𝑤 ∈ ω ↦ (𝐹‘𝑤))
38		ineq2 4166	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑤) → (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)))
39	38	eqeq1d 2738	. . . . . 6 ⊢ (𝑧 = (𝐹‘𝑤) → ((ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ 𝑧) = ∅ ↔ (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅))
40	37, 39	rexrnmptw 7045	. . . . 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 217	. . 3 ⊢ (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) → ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)
43		peano2 7827	. . . . . . . . 9 ⊢ (𝑤 ∈ ω → suc 𝑤 ∈ ω)
44	43	adantl 482	. . . . . . . 8 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → suc 𝑤 ∈ ω)
45		eqid 2736	. . . . . . . 8 ⊢ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)
46		fveq2 6842	. . . . . . . . 9 ⊢ (𝑦 = suc 𝑤 → (𝐹‘𝑦) = (𝐹‘suc 𝑤))
47	46	rspceeqv 3595	. . . . . . . 8 ⊢ ((suc 𝑤 ∈ ω ∧ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦))
48	44, 45, 47	sylancl 586	. . . . . . 7 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹‘𝑦))
49		fvex 6855	. . . . . . . 8 ⊢ (𝐹‘suc 𝑤) ∈ V
50	21	elrnmpt 5911	. . . . . . . 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 233	. . . . . 6 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)))
53		suceq 6383	. . . . . . . . . 10 ⊢ (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤)
54	53	fveq2d 6846	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑤))
55		fveq2 6842	. . . . . . . . 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 3580	. . . . . . 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 4424	. . . . . 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 584	. . . . 5 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) ≠ ∅)
61	60	neneqd 2948	. . . 4 ⊢ ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥)) ∧ 𝑤 ∈ ω) → ¬ (ran (𝑦 ∈ ω ↦ (𝐹‘𝑦)) ∩ (𝐹‘𝑤)) = ∅)
62	61	nrexdv 3146	. . 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 3103	. 2 ⊢ (∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥) ↔ ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))
65	63, 64	sylibr 233	1 ⊢ ((𝐹‘∅) ∈ On → ∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹‘𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3073  Vcvv 3445   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282   ↦ cmpt 5188   E cep 5536   We wwe 5587  dom cdm 5633  ran crn 5634  Oncon0 6317  suc csuc 6319  ⟶wf 6492  ‘cfv 6496  ωcom 7802

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-om 7803

This theorem is referenced by: (None)