Theorem noinfep 9125

Description: Using the Axiom of Regularity in the form zfregfr 9070, 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 9108	. . . . 5 ⊢ ω ∈ V
2	1	mptex 6988	. . . 4 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ∈ V
3	2	rnex 7619	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ∈ V
4		zfregfr 9070	. . 3 ⊢ E Fr ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))
5		ssid 3991	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ⊆ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))
6		dmmptg 6098	. . . . . 6 ⊢ (∀𝑤 ∈ ω (𝐹‘𝑤) ∈ V → dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ω)
7		fvexd 6687	. . . . . 6 ⊢ (𝑤 ∈ ω → (𝐹‘𝑤) ∈ V)
8	6, 7	mprg 3154	. . . . 5 ⊢ dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ω
9		peano1 7603	. . . . . 6 ⊢ ∅ ∈ ω
10	9	ne0ii 4305	. . . . 5 ⊢ ω ≠ ∅
11	8, 10	eqnetri 3088	. . . 4 ⊢ dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅
12		dm0rn0 5797	. . . . 5 ⊢ (dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ∅)
13	12	necon3bii 3070	. . . 4 ⊢ (dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅)
14	11, 13	mpbi 232	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅
15		fri 5519	. . 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 691	. 2 ⊢ ∃𝑦 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))∀𝑧 ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ¬ 𝑧 E 𝑦
17		fvex 6685	. . . . . . 7 ⊢ (𝐹‘𝑤) ∈ V
18		eqid 2823	. . . . . . 7 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = (𝑤 ∈ ω ↦ (𝐹‘𝑤))
19	17, 18	fnmpti 6493	. . . . . 6 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) Fn ω
20		fvelrnb 6728	. . . . . 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 7604	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
23		fveq2 6672	. . . . . . . . . . 11 ⊢ (𝑤 = suc 𝑥 → (𝐹‘𝑤) = (𝐹‘suc 𝑥))
24		fvex 6685	. . . . . . . . . . 11 ⊢ (𝐹‘suc 𝑥) ∈ V
25	23, 18, 24	fvmpt 6770	. . . . . . . . . 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 6850	. . . . . . . . . 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 589	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)))
29	26, 28	eqeltrrd 2916	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝐹‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)))
30		epel 5471	. . . . . . . . . . . 12 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
31		eleq1 2902	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (𝑧 ∈ 𝑦 ↔ (𝐹‘suc 𝑥) ∈ 𝑦))
32	30, 31	syl5bb 285	. . . . . . . . . . 11 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∈ 𝑦))
33	32	notbid 320	. . . . . . . . . 10 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ ¬ (𝐹‘suc 𝑥) ∈ 𝑦))
34		df-nel 3126	. . . . . . . . . 10 ⊢ ((𝐹‘suc 𝑥) ∉ 𝑦 ↔ ¬ (𝐹‘suc 𝑥) ∈ 𝑦)
35	33, 34	syl6bbr 291	. . . . . . . . 9 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∉ 𝑦))
36	35	rspccv 3622	. . . . . . . 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 6672	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
39		fvex 6685	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) ∈ V
40	38, 18, 39	fvmpt 6770	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = (𝐹‘𝑥))
41		eqeq1 2827	. . . . . . . . . 10 ⊢ (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = (𝐹‘𝑥) ↔ 𝑦 = (𝐹‘𝑥)))
42	40, 41	syl5ibcom 247	. . . . . . . . 9 ⊢ (𝑥 ∈ ω → (((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = 𝑦 → 𝑦 = (𝐹‘𝑥)))
43		neleq2 3131	. . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑥) → ((𝐹‘suc 𝑥) ∉ 𝑦 ↔ (𝐹‘suc 𝑥) ∉ (𝐹‘𝑥)))
44	43	biimpd 231	. . . . . . . . 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 3274	. . . . 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 244	. . . 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 3282	. 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 208   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   ∉ wnel 3125  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ⊆ wss 3938  ∅c0 4293   class class class wbr 5068   ↦ cmpt 5148   E cep 5466   Fr wfr 5513  dom cdm 5557  ran crn 5558  suc csuc 6195   Fn wfn 6352  ‘cfv 6357  ωcom 7582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463  ax-reg 9058  ax-inf2 9106

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583

This theorem is referenced by: (None)