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Theorem noinfep 9698
Description: Using the Axiom of Regularity in the form zfregfr 9643, show that there are no infinite descending -chains. Proposition 7.34 of [TakeutiZaring] p. 44. (Contributed by NM, 26-Jan-2006.) (Revised by Mario Carneiro, 22-Mar-2013.)
Assertion
Ref Expression
noinfep 𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹𝑥)
Distinct variable group:   𝑥,𝐹

Proof of Theorem noinfep
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 omex 9681 . . . . 5 ω ∈ V
21mptex 7243 . . . 4 (𝑤 ∈ ω ↦ (𝐹𝑤)) ∈ V
32rnex 7933 . . 3 ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ∈ V
4 zfregfr 9643 . . 3 E Fr ran (𝑤 ∈ ω ↦ (𝐹𝑤))
5 ssid 4018 . . 3 ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ⊆ ran (𝑤 ∈ ω ↦ (𝐹𝑤))
6 dmmptg 6264 . . . . . 6 (∀𝑤 ∈ ω (𝐹𝑤) ∈ V → dom (𝑤 ∈ ω ↦ (𝐹𝑤)) = ω)
7 fvexd 6922 . . . . . 6 (𝑤 ∈ ω → (𝐹𝑤) ∈ V)
86, 7mprg 3065 . . . . 5 dom (𝑤 ∈ ω ↦ (𝐹𝑤)) = ω
9 peano1 7911 . . . . . 6 ∅ ∈ ω
109ne0ii 4350 . . . . 5 ω ≠ ∅
118, 10eqnetri 3009 . . . 4 dom (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅
12 dm0rn0 5938 . . . . 5 (dom (𝑤 ∈ ω ↦ (𝐹𝑤)) = ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) = ∅)
1312necon3bii 2991 . . . 4 (dom (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅)
1411, 13mpbi 230 . . 3 ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅
15 fri 5646 . . 3 (((ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ∈ V ∧ E Fr ran (𝑤 ∈ ω ↦ (𝐹𝑤))) ∧ (ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ⊆ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ∧ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅)) → ∃𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤))∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ¬ 𝑧 E 𝑦)
163, 4, 5, 14, 15mp4an 693 . 2 𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤))∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ¬ 𝑧 E 𝑦
17 fvex 6920 . . . . . . 7 (𝐹𝑤) ∈ V
18 eqid 2735 . . . . . . 7 (𝑤 ∈ ω ↦ (𝐹𝑤)) = (𝑤 ∈ ω ↦ (𝐹𝑤))
1917, 18fnmpti 6712 . . . . . 6 (𝑤 ∈ ω ↦ (𝐹𝑤)) Fn ω
20 fvelrnb 6969 . . . . . 6 ((𝑤 ∈ ω ↦ (𝐹𝑤)) Fn ω → (𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ↔ ∃𝑥 ∈ ω ((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = 𝑦))
2119, 20ax-mp 5 . . . . 5 (𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ↔ ∃𝑥 ∈ ω ((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = 𝑦)
22 peano2 7913 . . . . . . . . . 10 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
23 fveq2 6907 . . . . . . . . . . 11 (𝑤 = suc 𝑥 → (𝐹𝑤) = (𝐹‘suc 𝑥))
24 fvex 6920 . . . . . . . . . . 11 (𝐹‘suc 𝑥) ∈ V
2523, 18, 24fvmpt 7016 . . . . . . . . . 10 (suc 𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹𝑤))‘suc 𝑥) = (𝐹‘suc 𝑥))
2622, 25syl 17 . . . . . . . . 9 (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹𝑤))‘suc 𝑥) = (𝐹‘suc 𝑥))
27 fnfvelrn 7100 . . . . . . . . . 10 (((𝑤 ∈ ω ↦ (𝐹𝑤)) Fn ω ∧ suc 𝑥 ∈ ω) → ((𝑤 ∈ ω ↦ (𝐹𝑤))‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)))
2819, 22, 27sylancr 587 . . . . . . . . 9 (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹𝑤))‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)))
2926, 28eqeltrrd 2840 . . . . . . . 8 (𝑥 ∈ ω → (𝐹‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)))
30 epel 5592 . . . . . . . . . . . 12 (𝑧 E 𝑦𝑧𝑦)
31 eleq1 2827 . . . . . . . . . . . 12 (𝑧 = (𝐹‘suc 𝑥) → (𝑧𝑦 ↔ (𝐹‘suc 𝑥) ∈ 𝑦))
3230, 31bitrid 283 . . . . . . . . . . 11 (𝑧 = (𝐹‘suc 𝑥) → (𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∈ 𝑦))
3332notbid 318 . . . . . . . . . 10 (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ ¬ (𝐹‘suc 𝑥) ∈ 𝑦))
34 df-nel 3045 . . . . . . . . . 10 ((𝐹‘suc 𝑥) ∉ 𝑦 ↔ ¬ (𝐹‘suc 𝑥) ∈ 𝑦)
3533, 34bitr4di 289 . . . . . . . . 9 (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∉ 𝑦))
3635rspccv 3619 . . . . . . . 8 (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ¬ 𝑧 E 𝑦 → ((𝐹‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) → (𝐹‘suc 𝑥) ∉ 𝑦))
3729, 36syl5com 31 . . . . . . 7 (𝑥 ∈ ω → (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ¬ 𝑧 E 𝑦 → (𝐹‘suc 𝑥) ∉ 𝑦))
38 fveq2 6907 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
39 fvex 6920 . . . . . . . . . . 11 (𝐹𝑥) ∈ V
4038, 18, 39fvmpt 7016 . . . . . . . . . 10 (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = (𝐹𝑥))
41 eqeq1 2739 . . . . . . . . . 10 (((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = 𝑦 → (((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = (𝐹𝑥) ↔ 𝑦 = (𝐹𝑥)))
4240, 41syl5ibcom 245 . . . . . . . . 9 (𝑥 ∈ ω → (((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = 𝑦𝑦 = (𝐹𝑥)))
43 neleq2 3051 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → ((𝐹‘suc 𝑥) ∉ 𝑦 ↔ (𝐹‘suc 𝑥) ∉ (𝐹𝑥)))
4443biimpd 229 . . . . . . . . 9 (𝑦 = (𝐹𝑥) → ((𝐹‘suc 𝑥) ∉ 𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹𝑥)))
4542, 44syl6 35 . . . . . . . 8 (𝑥 ∈ ω → (((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = 𝑦 → ((𝐹‘suc 𝑥) ∉ 𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹𝑥))))
4645com23 86 . . . . . . 7 (𝑥 ∈ ω → ((𝐹‘suc 𝑥) ∉ 𝑦 → (((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = 𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹𝑥))))
4737, 46syldc 48 . . . . . 6 (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ¬ 𝑧 E 𝑦 → (𝑥 ∈ ω → (((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = 𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹𝑥))))
4847reximdvai 3163 . . . . 5 (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ¬ 𝑧 E 𝑦 → (∃𝑥 ∈ ω ((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = 𝑦 → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹𝑥)))
4921, 48biimtrid 242 . . . 4 (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ¬ 𝑧 E 𝑦 → (𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹𝑥)))
5049com12 32 . . 3 (𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) → (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ¬ 𝑧 E 𝑦 → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹𝑥)))
5150rexlimiv 3146 . 2 (∃𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤))∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ¬ 𝑧 E 𝑦 → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹𝑥))
5216, 51ax-mp 5 1 𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206   = wceq 1537  wcel 2106  wne 2938  wnel 3044  wral 3059  wrex 3068  Vcvv 3478  wss 3963  c0 4339   class class class wbr 5148  cmpt 5231   E cep 5588   Fr wfr 5638  dom cdm 5689  ran crn 5690  suc csuc 6388   Fn wfn 6558  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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf2 9679
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-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  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-iun 4998  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-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888
This theorem is referenced by: (None)
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