Theorem noinfep 9629

Description: Using the Axiom of Regularity in the form zfregfr 9573, 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 9612	. . . . 5 ⊢ ω ∈ V
2	1	mptex 7222	. . . 4 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ∈ V
3	2	rnex 7907	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ∈ V
4		zfregfr 9573	. . 3 ⊢ E Fr ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))
5		ssid 3967	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ⊆ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))
6		dmmptg 6244	. . . . . 6 ⊢ (∀𝑤 ∈ ω (𝐹‘𝑤) ∈ V → dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ω)
7		fvexd 6897	. . . . . 6 ⊢ (𝑤 ∈ ω → (𝐹‘𝑤) ∈ V)
8	6, 7	mprg 3091	. . . . 5 ⊢ dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ω
9		peano1 7885	. . . . . 6 ⊢ ∅ ∈ ω
10	9	ne0ii 4305	. . . . 5 ⊢ ω ≠ ∅
11	8, 10	eqnetri 3034	. . . 4 ⊢ dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅
12		dm0rn0 5915	. . . . 5 ⊢ (dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ∅)
13	12	necon3bii 3016	. . . 4 ⊢ (dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅)
14	11, 13	mpbi 233	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅
15		fri 5620	. . 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 705	. 2 ⊢ ∃𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦
17		fvex 6895	. . . . . . 7 ⊢ (𝐹‘𝑤) ∈ V
18		eqid 2769	. . . . . . 7 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = (𝑤 ∈ ω ↦ (𝐹‘𝑤))
19	17, 18	fnmpti 6679	. . . . . 6 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) Fn ω
20		fvelrnb 6942	. . . . . 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 7886	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
23		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑤 = suc 𝑥 → (𝐹‘𝑤) = (𝐹‘suc 𝑥))
24		fvex 6895	. . . . . . . . . . 11 ⊢ (𝐹‘suc 𝑥) ∈ V
25	23, 18, 24	fvmpt 6990	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘suc 𝑥) = (𝐹‘suc 𝑥))
26	22, 25	syl 18	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘suc 𝑥) = (𝐹‘suc 𝑥))
27		fnfvelrn 7076	. . . . . . . . . 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 598	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)))
29	26, 28	eqeltrrd 2870	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝐹‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)))
30		epel 5565	. . . . . . . . . . . 12 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
31		eleq1 2857	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (𝑧 ∈ 𝑦 ↔ (𝐹‘suc 𝑥) ∈ 𝑦))
32	30, 31	bitrid 286	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∈ 𝑦))
33	32	notbid 321	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ ¬ (𝐹‘suc 𝑥) ∈ 𝑦))
34		df-nel 3071	. . . . . . . . . 10 ⊢ ((𝐹‘suc 𝑥) ∉ 𝑦 ↔ ¬ (𝐹‘suc 𝑥) ∈ 𝑦)
35	33, 34	bitr4di 292	. . . . . . . . 9 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∉ 𝑦))
36	35	rspccv 3587	. . . . . . . 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 32	. . . . . . 7 ⊢ (𝑥 ∈ ω → (∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦 → (𝐹‘suc 𝑥) ∉ 𝑦))
38		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
39		fvex 6895	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) ∈ V
40	38, 18, 39	fvmpt 6990	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = (𝐹‘𝑥))
41		eqeq1 2773	. . . . . . . . . 10 ⊢ (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = (𝐹‘𝑥) ↔ 𝑦 = (𝐹‘𝑥)))
42	40, 41	syl5ibcom 248	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → 𝑦 = (𝐹‘𝑥)))
43		neleq2 3077	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘suc 𝑥) ∉ 𝑦 ↔ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
44	43	biimpd 232	. . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘suc 𝑥) ∉ 𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
45	42, 44	syl6 36	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → ((𝐹‘suc 𝑥) ∉ 𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))))
46	45	com23 87	. . . . . . 7 ⊢ (𝑥 ∈ ω → ((𝐹‘suc 𝑥) ∉ 𝑦 → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥))))
47	37, 46	syldc 49	. . . . . 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 3182	. . . . 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 245	. . . 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 33	. . 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 3165	. 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 209   = wceq 1567   ∈ wcel 2149   ≠ wne 2964   ∉ wnel 3070  ∀wral 3085  ∃wrex 3095  Vcvv 3463   ⊆ wss 3913  ∅c0 4294   class class class wbr 5113   ↦ cmpt 5196   E cep 5561   Fr wfr 5612  dom cdm 5662  ran crn 5663  suc csuc 6363   Fn wfn 6532  ‘cfv 6537  ωcom 7862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733  ax-reg 9554  ax-inf2 9610

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7863

This theorem is referenced by: (None)