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Theorem noinfep 9601
Description: Using the Axiom of Regularity in the form zfregfr 9546, 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 9584 . . . . 5 ω ∈ V
21mptex 7174 . . . 4 (𝑤 ∈ ω ↦ (𝐹𝑤)) ∈ V
32rnex 7850 . . 3 ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ∈ V
4 zfregfr 9546 . . 3 E Fr ran (𝑤 ∈ ω ↦ (𝐹𝑤))
5 ssid 3967 . . 3 ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ⊆ ran (𝑤 ∈ ω ↦ (𝐹𝑤))
6 dmmptg 6195 . . . . . 6 (∀𝑤 ∈ ω (𝐹𝑤) ∈ V → dom (𝑤 ∈ ω ↦ (𝐹𝑤)) = ω)
7 fvexd 6858 . . . . . 6 (𝑤 ∈ ω → (𝐹𝑤) ∈ V)
86, 7mprg 3067 . . . . 5 dom (𝑤 ∈ ω ↦ (𝐹𝑤)) = ω
9 peano1 7826 . . . . . 6 ∅ ∈ ω
109ne0ii 4298 . . . . 5 ω ≠ ∅
118, 10eqnetri 3011 . . . 4 dom (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅
12 dm0rn0 5881 . . . . 5 (dom (𝑤 ∈ ω ↦ (𝐹𝑤)) = ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) = ∅)
1312necon3bii 2993 . . . 4 (dom (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅)
1411, 13mpbi 229 . . 3 ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅
15 fri 5594 . . 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 692 . 2 𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤))∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ¬ 𝑧 E 𝑦
17 fvex 6856 . . . . . . 7 (𝐹𝑤) ∈ V
18 eqid 2733 . . . . . . 7 (𝑤 ∈ ω ↦ (𝐹𝑤)) = (𝑤 ∈ ω ↦ (𝐹𝑤))
1917, 18fnmpti 6645 . . . . . 6 (𝑤 ∈ ω ↦ (𝐹𝑤)) Fn ω
20 fvelrnb 6904 . . . . . 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 7828 . . . . . . . . . 10 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
23 fveq2 6843 . . . . . . . . . . 11 (𝑤 = suc 𝑥 → (𝐹𝑤) = (𝐹‘suc 𝑥))
24 fvex 6856 . . . . . . . . . . 11 (𝐹‘suc 𝑥) ∈ V
2523, 18, 24fvmpt 6949 . . . . . . . . . 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 7032 . . . . . . . . . 10 (((𝑤 ∈ ω ↦ (𝐹𝑤)) Fn ω ∧ suc 𝑥 ∈ ω) → ((𝑤 ∈ ω ↦ (𝐹𝑤))‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)))
2819, 22, 27sylancr 588 . . . . . . . . 9 (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹𝑤))‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)))
2926, 28eqeltrrd 2835 . . . . . . . 8 (𝑥 ∈ ω → (𝐹‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)))
30 epel 5541 . . . . . . . . . . . 12 (𝑧 E 𝑦𝑧𝑦)
31 eleq1 2822 . . . . . . . . . . . 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 3047 . . . . . . . . . 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 3577 . . . . . . . 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 6843 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
39 fvex 6856 . . . . . . . . . . 11 (𝐹𝑥) ∈ V
4038, 18, 39fvmpt 6949 . . . . . . . . . 10 (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = (𝐹𝑥))
41 eqeq1 2737 . . . . . . . . . 10 (((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = 𝑦 → (((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = (𝐹𝑥) ↔ 𝑦 = (𝐹𝑥)))
4240, 41syl5ibcom 244 . . . . . . . . 9 (𝑥 ∈ ω → (((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = 𝑦𝑦 = (𝐹𝑥)))
43 neleq2 3052 . . . . . . . . . 10 (𝑦 = (𝐹𝑥) → ((𝐹‘suc 𝑥) ∉ 𝑦 ↔ (𝐹‘suc 𝑥) ∉ (𝐹𝑥)))
4443biimpd 228 . . . . . . . . 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 3159 . . . . 5 (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ¬ 𝑧 E 𝑦 → (∃𝑥 ∈ ω ((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = 𝑦 → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹𝑥)))
4921, 48biimtrid 241 . . . 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 3142 . 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 205   = wceq 1542  wcel 2107  wne 2940  wnel 3046  wral 3061  wrex 3070  Vcvv 3444  wss 3911  c0 4283   class class class wbr 5106  cmpt 5189   E cep 5537   Fr wfr 5586  dom cdm 5634  ran crn 5635  suc csuc 6320   Fn wfn 6492  cfv 6497  ωcom 7803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673  ax-reg 9533  ax-inf2 9582
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-om 7804
This theorem is referenced by: (None)
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