MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  onnseq Structured version   Visualization version   GIF version

Theorem onnseq 7301
Description: There are no length ω decreasing sequences in the ordinals. See also noinfep 8413 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 6848 . . . . . 6 E We On
21a1i 11 . . . . 5 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → E We On)
3 fveq2 6084 . . . . . . . . . . 11 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
43eleq1d 2667 . . . . . . . . . 10 (𝑦 = ∅ → ((𝐹𝑦) ∈ On ↔ (𝐹‘∅) ∈ On))
5 fveq2 6084 . . . . . . . . . . 11 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
65eleq1d 2667 . . . . . . . . . 10 (𝑦 = 𝑧 → ((𝐹𝑦) ∈ On ↔ (𝐹𝑧) ∈ On))
7 fveq2 6084 . . . . . . . . . . 11 (𝑦 = suc 𝑧 → (𝐹𝑦) = (𝐹‘suc 𝑧))
87eleq1d 2667 . . . . . . . . . 10 (𝑦 = suc 𝑧 → ((𝐹𝑦) ∈ On ↔ (𝐹‘suc 𝑧) ∈ On))
9 simpl 471 . . . . . . . . . 10 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → (𝐹‘∅) ∈ On)
10 suceq 5689 . . . . . . . . . . . . . . 15 (𝑥 = 𝑧 → suc 𝑥 = suc 𝑧)
1110fveq2d 6088 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑧))
12 fveq2 6084 . . . . . . . . . . . . . 14 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
1311, 12eleq12d 2677 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → ((𝐹‘suc 𝑥) ∈ (𝐹𝑥) ↔ (𝐹‘suc 𝑧) ∈ (𝐹𝑧)))
1413rspcv 3273 . . . . . . . . . . . 12 (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥) → (𝐹‘suc 𝑧) ∈ (𝐹𝑧)))
15 onelon 5647 . . . . . . . . . . . . 13 (((𝐹𝑧) ∈ On ∧ (𝐹‘suc 𝑧) ∈ (𝐹𝑧)) → (𝐹‘suc 𝑧) ∈ On)
1615expcom 449 . . . . . . . . . . . 12 ((𝐹‘suc 𝑧) ∈ (𝐹𝑧) → ((𝐹𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On))
1714, 16syl6 34 . . . . . . . . . . 11 (𝑧 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥) → ((𝐹𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On)))
1817adantld 481 . . . . . . . . . 10 (𝑧 ∈ ω → (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ((𝐹𝑧) ∈ On → (𝐹‘suc 𝑧) ∈ On)))
194, 6, 8, 9, 18finds2 6959 . . . . . . . . 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 2943 . . . . . . 7 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ∀𝑦 ∈ ω (𝐹𝑦) ∈ On)
22 eqid 2605 . . . . . . . 8 (𝑦 ∈ ω ↦ (𝐹𝑦)) = (𝑦 ∈ ω ↦ (𝐹𝑦))
2322fmpt 6270 . . . . . . 7 (∀𝑦 ∈ ω (𝐹𝑦) ∈ On ↔ (𝑦 ∈ ω ↦ (𝐹𝑦)):ω⟶On)
2421, 23sylib 206 . . . . . 6 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → (𝑦 ∈ ω ↦ (𝐹𝑦)):ω⟶On)
25 frn 5948 . . . . . 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 6950 . . . . . . . 8 ∅ ∈ ω
28 fdm 5946 . . . . . . . . 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 2690 . . . . . . 7 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ∅ ∈ dom (𝑦 ∈ ω ↦ (𝐹𝑦)))
31 ne0i 3875 . . . . . . 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 5246 . . . . . . 7 (dom (𝑦 ∈ ω ↦ (𝐹𝑦)) = ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) = ∅)
3433necon3bii 2829 . . . . . 6 (dom (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅ ↔ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅)
3532, 34sylib 206 . . . . 5 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ≠ ∅)
36 wefrc 5018 . . . . 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 1317 . . . 4 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ∃𝑧 ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦))(ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = ∅)
38 fvex 6094 . . . . . 6 (𝐹𝑤) ∈ V
3938rgenw 2903 . . . . 5 𝑤 ∈ ω (𝐹𝑤) ∈ V
40 fveq2 6084 . . . . . . 7 (𝑦 = 𝑤 → (𝐹𝑦) = (𝐹𝑤))
4140cbvmptv 4668 . . . . . 6 (𝑦 ∈ ω ↦ (𝐹𝑦)) = (𝑤 ∈ ω ↦ (𝐹𝑤))
42 ineq2 3765 . . . . . . 7 (𝑧 = (𝐹𝑤) → (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)))
4342eqeq1d 2607 . . . . . 6 (𝑧 = (𝐹𝑤) → ((ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ 𝑧) = ∅ ↔ (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅))
4441, 43rexrnmpt 6258 . . . . 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 206 . . 3 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅)
47 peano2 6951 . . . . . . . . 9 (𝑤 ∈ ω → suc 𝑤 ∈ ω)
4847adantl 480 . . . . . . . 8 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → suc 𝑤 ∈ ω)
49 eqid 2605 . . . . . . . 8 (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)
50 fveq2 6084 . . . . . . . . . 10 (𝑦 = suc 𝑤 → (𝐹𝑦) = (𝐹‘suc 𝑤))
5150eqeq2d 2615 . . . . . . . . 9 (𝑦 = suc 𝑤 → ((𝐹‘suc 𝑤) = (𝐹𝑦) ↔ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)))
5251rspcev 3277 . . . . . . . 8 ((suc 𝑤 ∈ ω ∧ (𝐹‘suc 𝑤) = (𝐹‘suc 𝑤)) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹𝑦))
5348, 49, 52sylancl 692 . . . . . . 7 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → ∃𝑦 ∈ ω (𝐹‘suc 𝑤) = (𝐹𝑦))
54 fvex 6094 . . . . . . . 8 (𝐹‘suc 𝑤) ∈ V
5522elrnmpt 5276 . . . . . . . 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 222 . . . . . 6 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦)))
58 suceq 5689 . . . . . . . . . 10 (𝑥 = 𝑤 → suc 𝑥 = suc 𝑤)
5958fveq2d 6088 . . . . . . . . 9 (𝑥 = 𝑤 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑤))
60 fveq2 6084 . . . . . . . . 9 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
6159, 60eleq12d 2677 . . . . . . . 8 (𝑥 = 𝑤 → ((𝐹‘suc 𝑥) ∈ (𝐹𝑥) ↔ (𝐹‘suc 𝑤) ∈ (𝐹𝑤)))
6261rspccva 3276 . . . . . . 7 ((∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹𝑤))
6362adantll 745 . . . . . 6 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → (𝐹‘suc 𝑤) ∈ (𝐹𝑤))
64 inelcm 3979 . . . . . 6 (((𝐹‘suc 𝑤) ∈ ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∧ (𝐹‘suc 𝑤) ∈ (𝐹𝑤)) → (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) ≠ ∅)
6557, 63, 64syl2anc 690 . . . . 5 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) ≠ ∅)
6665neneqd 2782 . . . 4 ((((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) ∧ 𝑤 ∈ ω) → ¬ (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅)
6766nrexdv 2979 . . 3 (((𝐹‘∅) ∈ On ∧ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥)) → ¬ ∃𝑤 ∈ ω (ran (𝑦 ∈ ω ↦ (𝐹𝑦)) ∩ (𝐹𝑤)) = ∅)
6846, 67pm2.65da 597 . 2 ((𝐹‘∅) ∈ On → ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥))
69 rexnal 2973 . 2 (∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹𝑥) ↔ ¬ ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ∈ (𝐹𝑥))
7068, 69sylibr 222 1 ((𝐹‘∅) ∈ On → ∃𝑥 ∈ ω ¬ (𝐹‘suc 𝑥) ∈ (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382   = wceq 1474  wcel 1975  wne 2775  wral 2891  wrex 2892  Vcvv 3168  cin 3534  wss 3535  c0 3869  cmpt 4633   E cep 4933   We wwe 4982  dom cdm 5024  ran crn 5025  Oncon0 5622  suc csuc 5624  wf 5782  cfv 5786  ωcom 6930
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-pss 3551  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-tp 4125  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-tr 4671  df-eprel 4935  df-id 4939  df-po 4945  df-so 4946  df-fr 4983  df-we 4985  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-ord 5625  df-on 5626  df-lim 5627  df-suc 5628  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-fv 5794  df-om 6931
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator