Theorem noinfep 9572

Description: Using the Axiom of Regularity in the form zfregfr 9516, 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) ∉ (𝐹‘𝑥)

Distinct variable group:   𝑥,𝐹

Proof of Theorem noinfep

Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omex 9555	. . . . 5 ⊢ ω ∈ V
2	1	mptex 7167	. . . 4 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ∈ V
3	2	rnex 7850	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ∈ V
4		zfregfr 9516	. . 3 ⊢ E Fr ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))
5		ssid 3937	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ⊆ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))
6		dmmptg 6193	. . . . . 6 ⊢ (∀𝑤 ∈ ω (𝐹‘𝑤) ∈ V → dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ω)
7		fvexd 6842	. . . . . 6 ⊢ (𝑤 ∈ ω → (𝐹‘𝑤) ∈ V)
8	6, 7	mprg 3059	. . . . 5 ⊢ dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ω
9		peano1 7829	. . . . . 6 ⊢ ∅ ∈ ω
10	9	ne0ii 4272	. . . . 5 ⊢ ω ≠ ∅
11	8, 10	eqnetri 3004	. . . 4 ⊢ dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅
12		dm0rn0 5866	. . . . 5 ⊢ (dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ∅)
13	12	necon3bii 2986	. . . 4 ⊢ (dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅)
14	11, 13	mpbi 231	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅
15		fri 5576	. . 3 ⊢ (((ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ∈ V ∧ E Fr ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))) ∧ (ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ⊆ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ∧ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅)) → ∃𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦)
16	3, 4, 5, 14, 15	mp4an 699	. 2 ⊢ ∃𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦
17		fvex 6840	. . . . . . 7 ⊢ (𝐹‘𝑤) ∈ V
18		eqid 2739	. . . . . . 7 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = (𝑤 ∈ ω ↦ (𝐹‘𝑤))
19	17, 18	fnmpti 6628	. . . . . 6 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) Fn ω
20		fvelrnb 6887	. . . . . 6 ⊢ ((𝑤 ∈ ω ↦ (𝐹‘𝑤)) Fn ω → (𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ↔ ∃𝑥 ∈ ω ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦))
21	19, 20	ax-mp 5	. . . . 5 ⊢ (𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ↔ ∃𝑥 ∈ ω ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦)
22		peano2 7830	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
23		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑤 = suc 𝑥 → (𝐹‘𝑤) = (𝐹‘suc 𝑥))
24		fvex 6840	. . . . . . . . . . 11 ⊢ (𝐹‘suc 𝑥) ∈ V
25	23, 18, 24	fvmpt 6935	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘suc 𝑥) = (𝐹‘suc 𝑥))
26	22, 25	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘suc 𝑥) = (𝐹‘suc 𝑥))
27		fnfvelrn 7021	. . . . . . . . . 10 ⊢ (((𝑤 ∈ ω ↦ (𝐹‘𝑤)) Fn ω ∧ suc 𝑥 ∈ ω) → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)))
28	19, 22, 27	sylancr 593	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)))
29	26, 28	eqeltrrd 2840	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝐹‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)))
30		epel 5521	. . . . . . . . . . . 12 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
31		eleq1 2827	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (𝑧 ∈ 𝑦 ↔ (𝐹‘suc 𝑥) ∈ 𝑦))
32	30, 31	bitrid 284	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∈ 𝑦))
33	32	notbid 319	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ ¬ (𝐹‘suc 𝑥) ∈ 𝑦))
34		df-nel 3039	. . . . . . . . . 10 ⊢ ((𝐹‘suc 𝑥) ∉ 𝑦 ↔ ¬ (𝐹‘suc 𝑥) ∈ 𝑦)
35	33, 34	bitr4di 290	. . . . . . . . 9 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∉ 𝑦))
36	35	rspccv 3557	. . . . . . . 8 ⊢ (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦 → ((𝐹‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) → (𝐹‘suc 𝑥) ∉ 𝑦))
37	29, 36	syl5com 31	. . . . . . 7 ⊢ (𝑥 ∈ ω → (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦 → (𝐹‘suc 𝑥) ∉ 𝑦))
38		fveq2 6827	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
39		fvex 6840	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) ∈ V
40	38, 18, 39	fvmpt 6935	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = (𝐹‘𝑥))
41		eqeq1 2743	. . . . . . . . . 10 ⊢ (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = (𝐹‘𝑥) ↔ 𝑦 = (𝐹‘𝑥)))
42	40, 41	syl5ibcom 246	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → 𝑦 = (𝐹‘𝑥)))
43		neleq2 3045	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘suc 𝑥) ∉ 𝑦 ↔ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
44	43	biimpd 230	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘suc 𝑥) ∉ 𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
45	42, 44	syl6 35	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → ((𝐹‘suc 𝑥) ∉ 𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))))
46	45	com23 86	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((𝐹‘suc 𝑥) ∉ 𝑦 → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))))
47	37, 46	syldc 48	. . . . . 6 ⊢ (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦 → (𝑥 ∈ ω → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))))
48	47	reximdvai 3150	. . . . 5 ⊢ (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦 → (∃𝑥 ∈ ω ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
49	21, 48	biimtrid 243	. . . 4 ⊢ (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦 → (𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
50	49	com12 32	. . 3 ⊢ (𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) → (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦 → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
51	50	rexlimiv 3133	. 2 ⊢ (∃𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦 → ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))
52	16, 51	ax-mp 5	1 ⊢ ∃𝑥 ∈ ω (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   ∉ wnel 3038  ∀wral 3053  ∃wrex 3063  Vcvv 3431   ⊆ wss 3883  ∅c0 4261   class class class wbr 5072   ↦ cmpt 5153   E cep 5517   Fr wfr 5568  dom cdm 5618  ran crn 5619  suc csuc 6312   Fn wfn 6480  ‘cfv 6485  ωcom 7806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678  ax-reg 9497  ax-inf2 9553

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-om 7807

This theorem is referenced by: (None)