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Theorem onnseq 7593
Description: There are no length ω decreasing sequences in the ordinals. See also noinfep 8720 for a stronger version assuming Regularity. (Contributed by Mario Carneiro, 19-May-2015.)
Assertion
Ref Expression
onnseq ((𝐹‘∅) ∈ On → ∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹𝑥))
Distinct variable group:   𝑥,𝐹

Proof of Theorem onnseq
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 epweon 7129 . . . . . 6 E We On
21a1i 11 . . . . 5 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → E We On)
3 fveq2 6332 . . . . . . . . . . 11 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
43eleq1d 2834 . . . . . . . . . 10 (𝑦 = ∅ → ((𝐹𝑦) ∈ On ↔ (𝐹‘∅) ∈ On))
5 fveq2 6332 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
65eleq1d 2834 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝐹𝑦) ∈ On ↔ (𝐹𝑧) ∈ On))
7 fveq2 6332 . . . . . . . . . . 11 (𝑦 = suc 𝑧 → (𝐹𝑦) = (𝐹‘suc 𝑧))
87eleq1d 2834 . . . . . . . . . 10 (𝑦 = suc 𝑧 → ((𝐹𝑦) ∈ On ↔ (𝐹‘suc 𝑧) ∈ On))
9 simpl 468 . . . . . . . . . 10 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → (𝐹‘∅) ∈ On)
10 suceq 5933 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
1110fveq2d 6336 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑧))
12 fveq2 6332 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
1311, 12eleq12d 2843 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹‘suc 𝑥) ∈ (𝐹𝑥) ↔ (𝐹‘suc 𝑧) ∈ (𝐹𝑧)))
1413rspcv 3454 . . . . . . . . . . . 12 (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥) → (𝐹‘suc 𝑧) ∈ (𝐹𝑧)))
15 onelon 5891 . . . . . . . . . . . . 13 (((𝐹𝑧) ∈ On ∧ (𝐹‘suc 𝑧) ∈ (𝐹𝑧)) → (𝐹‘suc 𝑧) ∈ On)
1615expcom 398 . . . . . . . . . . . 12 ((𝐹‘suc 𝑧) ∈ (𝐹𝑧) → ((𝐹𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On))
1714, 16syl6 35 . . . . . . . . . . 11 (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥) → ((𝐹𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On)))
1817adantld 474 . . . . . . . . . 10 (𝑧 ∈ ω → (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ((𝐹𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On)))
194, 6, 8, 9, 18finds2 7240 . . . . . . . . 9 (𝑦 ∈ ω → (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → (𝐹𝑦) ∈ On))
2019com12 32 . . . . . . . 8 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → (𝑦 ∈ ω → (𝐹𝑦) ∈ On))
2120ralrimiv 3113 . . . . . . 7 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ∀𝑦 ∈ ω (𝐹𝑦) ∈ On)
22 eqid 2770 . . . . . . . 8 (𝑦 ∈ ω ↦ (𝐹𝑦)) = (𝑦 ∈ ω ↦ (𝐹𝑦))
2322fmpt 6523 . . . . . . 7 (∀𝑦 ∈ ω (𝐹𝑦) ∈ On ↔ (𝑦 ∈ ω ↦ (𝐹𝑦)):ω⟶On)
2421, 23sylib 208 . . . . . 6 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → (𝑦 ∈ ω ↦ (𝐹𝑦)):ω⟶On)
25 frn 6193 . . . . . 6 ((𝑦 ∈ ω ↦ (𝐹𝑦)):ω⟶On → ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ⊆ On)
2624, 25syl 17 . . . . 5 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ⊆ On)
27 peano1 7231 . . . . . . . 8 ∅ ∈ ω
28 fdm 6191 . . . . . . . . 9 ((𝑦 ∈ ω ↦ (𝐹𝑦)):ω⟶On → dom (𝑦 ∈ ω ↦ (𝐹𝑦)) = ω)
2924, 28syl 17 . . . . . . . 8 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹𝑦)) = ω)
3027, 29syl5eleqr 2856 . . . . . . 7 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ∅ ∈ dom (𝑦 ∈ ω ↦ (𝐹𝑦)))
31 ne0i 4067 . . . . . . 7 (∅ ∈ dom (𝑦 ∈ ω ↦ (𝐹𝑦)) → dom (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅)
3230, 31syl 17 . . . . . 6 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅)
33 dm0rn0 5480 . . . . . . 7 (dom (𝑦 ∈ ω ↦ (𝐹𝑦)) = ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) = ∅)
3433necon3bii 2994 . . . . . 6 (dom (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅)
3532, 34sylib 208 . . . . 5 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅)
36 wefrc 5243 . . . . 5 (( E We On ∧ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ⊆ On ∧ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅) → ∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦))(ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = ∅)
372, 26, 35, 36syl3anc 1475 . . . 4 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦))(ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = ∅)
38 fvex 6342 . . . . . 6 (𝐹𝑤) ∈ V
3938rgenw 3072 . . . . 5 𝑤 ∈ ω (𝐹𝑤) ∈ V
40 fveq2 6332 . . . . . . 7 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
4140cbvmptv 4882 . . . . . 6 (𝑦 ∈ ω ↦ (𝐹𝑦)) = (𝑤 ∈ ω ↦ (𝐹𝑤))
42 ineq2 3957 . . . . . . 7 (𝑧 = (𝐹𝑤) → (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)))
4342eqeq1d 2772 . . . . . 6 (𝑧 = (𝐹𝑤) → ((ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = ∅ ↔ (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅))
4441, 43rexrnmpt 6512 . . . . 5 (∀𝑤 ∈ ω (𝐹𝑤) ∈ V → (∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦))(ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = ∅ ↔ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅))
4539, 44ax-mp 5 . . . 4 (∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦))(ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = ∅ ↔ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅)
4637, 45sylib 208 . . 3 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅)
47 peano2 7232 . . . . . . . . 9 (𝑤 ∈ ω → suc 𝑤 ∈ ω)
4847adantl 467 . . . . . . . 8 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → suc 𝑤 ∈ ω)
49 eqid 2770 . . . . . . . 8 (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)
50 fveq2 6332 . . . . . . . . . 10 (𝑦 = suc 𝑤 → (𝐹𝑦) = (𝐹‘suc 𝑤))
5150eqeq2d 2780 . . . . . . . . 9 (𝑦 = suc 𝑤 → ((𝐹‘suc 𝑤) = (𝐹𝑦) ↔ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)))
5251rspcev 3458 . . . . . . . 8 ((suc 𝑤 ∈ ω ∧ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹𝑦))
5348, 49, 52sylancl 566 . . . . . . 7 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹𝑦))
54 fvex 6342 . . . . . . . 8 (𝐹‘suc 𝑤) ∈ V
5522elrnmpt 5510 . . . . . . . 8 ((𝐹‘suc 𝑤) ∈ V → ((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ↔ ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹𝑦)))
5654, 55ax-mp 5 . . . . . . 7 ((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ↔ ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹𝑦))
5753, 56sylibr 224 . . . . . 6 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦)))
58 suceq 5933 . . . . . . . . . 10 (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤)
5958fveq2d 6336 . . . . . . . . 9 (𝑥 = 𝑤 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑤))
60 fveq2 6332 . . . . . . . . 9 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
6159, 60eleq12d 2843 . . . . . . . 8 (𝑥 = 𝑤 → ((𝐹‘suc 𝑥) ∈ (𝐹𝑥) ↔ (𝐹‘suc 𝑤) ∈ (𝐹𝑤)))
6261rspccva 3457 . . . . . . 7 ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹𝑤))
6362adantll 685 . . . . . 6 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹𝑤))
64 inelcm 4173 . . . . . 6 (((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∧ (𝐹‘suc 𝑤) ∈ (𝐹𝑤)) → (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) ≠ ∅)
6557, 63, 64syl2anc 565 . . . . 5 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) ≠ ∅)
6665neneqd 2947 . . . 4 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → ¬ (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅)
6766nrexdv 3148 . . 3 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ¬ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅)
6846, 67pm2.65da 800 . 2 ((𝐹‘∅) ∈ On → ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥))
69 rexnal 3142 . 2 (∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹𝑥) ↔ ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥))
7068, 69sylibr 224 1 ((𝐹‘∅) ∈ On → ∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wne 2942  wral 3060  wrex 3061  Vcvv 3349  cin 3720  wss 3721  c0 4061  cmpt 4861   E cep 5161   We wwe 5207  dom cdm 5249  ran crn 5250  Oncon0 5866  suc csuc 5868  wf 6027  cfv 6031  ωcom 7211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-om 7212
This theorem is referenced by: (None)
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