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Theorem noinfep 9596
Description: Using the Axiom of Regularity in the form zfregfr 9541, 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 9579 . . . . 5 ω ∈ V
21mptex 7173 . . . 4 (𝑤 ∈ ω ↦ (𝐹𝑤)) ∈ V
32rnex 7849 . . 3 ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ∈ V
4 zfregfr 9541 . . 3 E Fr ran (𝑤 ∈ ω ↦ (𝐹𝑤))
5 ssid 3966 . . 3 ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ⊆ ran (𝑤 ∈ ω ↦ (𝐹𝑤))
6 dmmptg 6194 . . . . . 6 (∀𝑤 ∈ ω (𝐹𝑤) ∈ V → dom (𝑤 ∈ ω ↦ (𝐹𝑤)) = ω)
7 fvexd 6857 . . . . . 6 (𝑤 ∈ ω → (𝐹𝑤) ∈ V)
86, 7mprg 3070 . . . . 5 dom (𝑤 ∈ ω ↦ (𝐹𝑤)) = ω
9 peano1 7825 . . . . . 6 ∅ ∈ ω
109ne0ii 4297 . . . . 5 ω ≠ ∅
118, 10eqnetri 3014 . . . 4 dom (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅
12 dm0rn0 5880 . . . . 5 (dom (𝑤 ∈ ω ↦ (𝐹𝑤)) = ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) = ∅)
1312necon3bii 2996 . . . 4 (dom (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅)
1411, 13mpbi 229 . . 3 ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ≠ ∅
15 fri 5593 . . 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 691 . 2 𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤))∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)) ¬ 𝑧 E 𝑦
17 fvex 6855 . . . . . . 7 (𝐹𝑤) ∈ V
18 eqid 2736 . . . . . . 7 (𝑤 ∈ ω ↦ (𝐹𝑤)) = (𝑤 ∈ ω ↦ (𝐹𝑤))
1917, 18fnmpti 6644 . . . . . 6 (𝑤 ∈ ω ↦ (𝐹𝑤)) Fn ω
20 fvelrnb 6903 . . . . . 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 7827 . . . . . . . . . 10 (𝑥 ∈ ω → suc 𝑥 ∈ ω)
23 fveq2 6842 . . . . . . . . . . 11 (𝑤 = suc 𝑥 → (𝐹𝑤) = (𝐹‘suc 𝑥))
24 fvex 6855 . . . . . . . . . . 11 (𝐹‘suc 𝑥) ∈ V
2523, 18, 24fvmpt 6948 . . . . . . . . . 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 7031 . . . . . . . . . 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 2839 . . . . . . . 8 (𝑥 ∈ ω → (𝐹‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹𝑤)))
30 epel 5540 . . . . . . . . . . . 12 (𝑧 E 𝑦𝑧𝑦)
31 eleq1 2825 . . . . . . . . . . . 12 (𝑧 = (𝐹‘suc 𝑥) → (𝑧𝑦 ↔ (𝐹‘suc 𝑥) ∈ 𝑦))
3230, 31bitrid 282 . . . . . . . . . . 11 (𝑧 = (𝐹‘suc 𝑥) → (𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∈ 𝑦))
3332notbid 317 . . . . . . . . . 10 (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ ¬ (𝐹‘suc 𝑥) ∈ 𝑦))
34 df-nel 3050 . . . . . . . . . 10 ((𝐹‘suc 𝑥) ∉ 𝑦 ↔ ¬ (𝐹‘suc 𝑥) ∈ 𝑦)
3533, 34bitr4di 288 . . . . . . . . 9 (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∉ 𝑦))
3635rspccv 3578 . . . . . . . 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 6842 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝐹𝑤) = (𝐹𝑥))
39 fvex 6855 . . . . . . . . . . 11 (𝐹𝑥) ∈ V
4038, 18, 39fvmpt 6948 . . . . . . . . . 10 (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹𝑤))‘𝑥) = (𝐹𝑥))
41 eqeq1 2740 . . . . . . . . . 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 3055 . . . . . . . . . 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 3162 . . . . 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 3145 . 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 1541  wcel 2106  wne 2943  wnel 3049  wral 3064  wrex 3073  Vcvv 3445  wss 3910  c0 4282   class class class wbr 5105  cmpt 5188   E cep 5536   Fr wfr 5585  dom cdm 5633  ran crn 5634  suc csuc 6319   Fn wfn 6491  cfv 6496  ωcom 7802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672  ax-reg 9528  ax-inf2 9577
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803
This theorem is referenced by: (None)
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