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Theorem onnseq 8383
Description: There are no length ω decreasing sequences in the ordinals. See also noinfep 9698 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 7794 . . . . 5 E We On
2 fveq2 6907 . . . . . . . . . . 11 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
32eleq1d 2824 . . . . . . . . . 10 (𝑦 = ∅ → ((𝐹𝑦) ∈ On ↔ (𝐹‘∅) ∈ On))
4 fveq2 6907 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
54eleq1d 2824 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝐹𝑦) ∈ On ↔ (𝐹𝑧) ∈ On))
6 fveq2 6907 . . . . . . . . . . 11 (𝑦 = suc 𝑧 → (𝐹𝑦) = (𝐹‘suc 𝑧))
76eleq1d 2824 . . . . . . . . . 10 (𝑦 = suc 𝑧 → ((𝐹𝑦) ∈ On ↔ (𝐹‘suc 𝑧) ∈ On))
8 simpl 482 . . . . . . . . . 10 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → (𝐹‘∅) ∈ On)
9 suceq 6452 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
109fveq2d 6911 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑧))
11 fveq2 6907 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
1210, 11eleq12d 2833 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹‘suc 𝑥) ∈ (𝐹𝑥) ↔ (𝐹‘suc 𝑧) ∈ (𝐹𝑧)))
1312rspcv 3618 . . . . . . . . . . . 12 (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥) → (𝐹‘suc 𝑧) ∈ (𝐹𝑧)))
14 onelon 6411 . . . . . . . . . . . . 13 (((𝐹𝑧) ∈ On ∧ (𝐹‘suc 𝑧) ∈ (𝐹𝑧)) → (𝐹‘suc 𝑧) ∈ On)
1514expcom 413 . . . . . . . . . . . 12 ((𝐹‘suc 𝑧) ∈ (𝐹𝑧) → ((𝐹𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On))
1613, 15syl6 35 . . . . . . . . . . 11 (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥) → ((𝐹𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On)))
1716adantld 490 . . . . . . . . . 10 (𝑧 ∈ ω → (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ((𝐹𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On)))
183, 5, 7, 8, 17finds2 7921 . . . . . . . . 9 (𝑦 ∈ ω → (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → (𝐹𝑦) ∈ On))
1918com12 32 . . . . . . . 8 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → (𝑦 ∈ ω → (𝐹𝑦) ∈ On))
2019ralrimiv 3143 . . . . . . 7 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ∀𝑦 ∈ ω (𝐹𝑦) ∈ On)
21 eqid 2735 . . . . . . . 8 (𝑦 ∈ ω ↦ (𝐹𝑦)) = (𝑦 ∈ ω ↦ (𝐹𝑦))
2221fmpt 7130 . . . . . . 7 (∀𝑦 ∈ ω (𝐹𝑦) ∈ On ↔ (𝑦 ∈ ω ↦ (𝐹𝑦)):ω⟶On)
2320, 22sylib 218 . . . . . 6 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → (𝑦 ∈ ω ↦ (𝐹𝑦)):ω⟶On)
2423frnd 6745 . . . . 5 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ⊆ On)
25 peano1 7911 . . . . . . . 8 ∅ ∈ ω
2623fdmd 6747 . . . . . . . 8 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹𝑦)) = ω)
2725, 26eleqtrrid 2846 . . . . . . 7 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ∅ ∈ dom (𝑦 ∈ ω ↦ (𝐹𝑦)))
2827ne0d 4348 . . . . . 6 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → dom (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅)
29 dm0rn0 5938 . . . . . . 7 (dom (𝑦 ∈ ω ↦ (𝐹𝑦)) = ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) = ∅)
3029necon3bii 2991 . . . . . 6 (dom (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅)
3128, 30sylib 218 . . . . 5 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅)
32 wefrc 5683 . . . . 5 (( E We On ∧ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ⊆ On ∧ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅) → ∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦))(ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = ∅)
331, 24, 31, 32mp3an2i 1465 . . . 4 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦))(ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = ∅)
34 fvex 6920 . . . . . 6 (𝐹𝑤) ∈ V
3534rgenw 3063 . . . . 5 𝑤 ∈ ω (𝐹𝑤) ∈ V
36 fveq2 6907 . . . . . . 7 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
3736cbvmptv 5261 . . . . . 6 (𝑦 ∈ ω ↦ (𝐹𝑦)) = (𝑤 ∈ ω ↦ (𝐹𝑤))
38 ineq2 4222 . . . . . . 7 (𝑧 = (𝐹𝑤) → (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)))
3938eqeq1d 2737 . . . . . 6 (𝑧 = (𝐹𝑤) → ((ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = ∅ ↔ (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅))
4037, 39rexrnmptw 7115 . . . . 5 (∀𝑤 ∈ ω (𝐹𝑤) ∈ V → (∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦))(ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = ∅ ↔ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅))
4135, 40ax-mp 5 . . . 4 (∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦))(ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = ∅ ↔ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅)
4233, 41sylib 218 . . 3 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅)
43 peano2 7913 . . . . . . . . 9 (𝑤 ∈ ω → suc 𝑤 ∈ ω)
4443adantl 481 . . . . . . . 8 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → suc 𝑤 ∈ ω)
45 eqid 2735 . . . . . . . 8 (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)
46 fveq2 6907 . . . . . . . . 9 (𝑦 = suc 𝑤 → (𝐹𝑦) = (𝐹‘suc 𝑤))
4746rspceeqv 3645 . . . . . . . 8 ((suc 𝑤 ∈ ω ∧ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹𝑦))
4844, 45, 47sylancl 586 . . . . . . 7 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹𝑦))
49 fvex 6920 . . . . . . . 8 (𝐹‘suc 𝑤) ∈ V
5021elrnmpt 5972 . . . . . . . 8 ((𝐹‘suc 𝑤) ∈ V → ((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ↔ ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹𝑦)))
5149, 50ax-mp 5 . . . . . . 7 ((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ↔ ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹𝑦))
5248, 51sylibr 234 . . . . . 6 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦)))
53 suceq 6452 . . . . . . . . . 10 (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤)
5453fveq2d 6911 . . . . . . . . 9 (𝑥 = 𝑤 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑤))
55 fveq2 6907 . . . . . . . . 9 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
5654, 55eleq12d 2833 . . . . . . . 8 (𝑥 = 𝑤 → ((𝐹‘suc 𝑥) ∈ (𝐹𝑥) ↔ (𝐹‘suc 𝑤) ∈ (𝐹𝑤)))
5756rspccva 3621 . . . . . . 7 ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹𝑤))
5857adantll 714 . . . . . 6 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹𝑤))
59 inelcm 4471 . . . . . 6 (((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∧ (𝐹‘suc 𝑤) ∈ (𝐹𝑤)) → (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) ≠ ∅)
6052, 58, 59syl2anc 584 . . . . 5 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) ≠ ∅)
6160neneqd 2943 . . . 4 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → ¬ (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅)
6261nrexdv 3147 . . 3 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ¬ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅)
6342, 62pm2.65da 817 . 2 ((𝐹‘∅) ∈ On → ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥))
64 rexnal 3098 . 2 (∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹𝑥) ↔ ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥))
6563, 64sylibr 234 1 ((𝐹‘∅) ∈ On → ∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  Vcvv 3478  cin 3962  wss 3963  c0 4339  cmpt 5231   E cep 5588   We wwe 5640  dom cdm 5689  ran crn 5690  Oncon0 6386  suc csuc 6388  wf 6559  cfv 6563  ωcom 7887
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-om 7888
This theorem is referenced by: (None)
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