Theorem noinfep 9396

Description: Using the Axiom of Regularity in the form zfregfr 9341, 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 9379	. . . . 5 ⊢ ω ∈ V
2	1	mptex 7096	. . . 4 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ∈ V
3	2	rnex 7753	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ∈ V
4		zfregfr 9341	. . 3 ⊢ E Fr ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))
5		ssid 3948	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ⊆ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))
6		dmmptg 6144	. . . . . 6 ⊢ (∀𝑤 ∈ ω (𝐹‘𝑤) ∈ V → dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ω)
7		fvexd 6786	. . . . . 6 ⊢ (𝑤 ∈ ω → (𝐹‘𝑤) ∈ V)
8	6, 7	mprg 3080	. . . . 5 ⊢ dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ω
9		peano1 7729	. . . . . 6 ⊢ ∅ ∈ ω
10	9	ne0ii 4277	. . . . 5 ⊢ ω ≠ ∅
11	8, 10	eqnetri 3016	. . . 4 ⊢ dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅
12		dm0rn0 5833	. . . . 5 ⊢ (dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ∅)
13	12	necon3bii 2998	. . . 4 ⊢ (dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅)
14	11, 13	mpbi 229	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅
15		fri 5550	. . 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 690	. 2 ⊢ ∃𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦
17		fvex 6784	. . . . . . 7 ⊢ (𝐹‘𝑤) ∈ V
18		eqid 2740	. . . . . . 7 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = (𝑤 ∈ ω ↦ (𝐹‘𝑤))
19	17, 18	fnmpti 6574	. . . . . 6 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) Fn ω
20		fvelrnb 6827	. . . . . 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 7731	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
23		fveq2 6771	. . . . . . . . . . 11 ⊢ (𝑤 = suc 𝑥 → (𝐹‘𝑤) = (𝐹‘suc 𝑥))
24		fvex 6784	. . . . . . . . . . 11 ⊢ (𝐹‘suc 𝑥) ∈ V
25	23, 18, 24	fvmpt 6872	. . . . . . . . . 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 6955	. . . . . . . . . 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 587	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)))
29	26, 28	eqeltrrd 2842	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝐹‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)))
30		epel 5499	. . . . . . . . . . . 12 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
31		eleq1 2828	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (𝑧 ∈ 𝑦 ↔ (𝐹‘suc 𝑥) ∈ 𝑦))
32	30, 31	bitrid 282	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∈ 𝑦))
33	32	notbid 318	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ ¬ (𝐹‘suc 𝑥) ∈ 𝑦))
34		df-nel 3052	. . . . . . . . . 10 ⊢ ((𝐹‘suc 𝑥) ∉ 𝑦 ↔ ¬ (𝐹‘suc 𝑥) ∈ 𝑦)
35	33, 34	bitr4di 289	. . . . . . . . 9 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∉ 𝑦))
36	35	rspccv 3558	. . . . . . . 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 6771	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
39		fvex 6784	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) ∈ V
40	38, 18, 39	fvmpt 6872	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = (𝐹‘𝑥))
41		eqeq1 2744	. . . . . . . . . 10 ⊢ (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = (𝐹‘𝑥) ↔ 𝑦 = (𝐹‘𝑥)))
42	40, 41	syl5ibcom 244	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → 𝑦 = (𝐹‘𝑥)))
43		neleq2 3057	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘suc 𝑥) ∉ 𝑦 ↔ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
44	43	biimpd 228	. . . . . . . . 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 3202	. . . . 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	syl5bi 241	. . . 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 3211	. 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 205   = wceq 1542   ∈ wcel 2110   ≠ wne 2945   ∉ wnel 3051  ∀wral 3066  ∃wrex 3067  Vcvv 3431   ⊆ wss 3892  ∅c0 4262   class class class wbr 5079   ↦ cmpt 5162   E cep 5495   Fr wfr 5542  dom cdm 5590  ran crn 5591  suc csuc 6267   Fn wfn 6427  ‘cfv 6432  ωcom 7706

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 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582  ax-reg 9329  ax-inf2 9377

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-om 7707

This theorem is referenced by: (None)