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) ∉ (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 9579	. . . . 5 ⊢ ω ∈ V
2	1	mptex 7173	. . . 4 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ∈ V
3	2	rnex 7849	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ∈ V
4		zfregfr 9541	. . 3 ⊢ E Fr ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))
5		ssid 3966	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ⊆ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤))
6		dmmptg 6194	. . . . . 6 ⊢ (∀𝑤 ∈ ω (𝐹‘𝑤) ∈ V → dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ω)
7		fvexd 6857	. . . . . 6 ⊢ (𝑤 ∈ ω → (𝐹‘𝑤) ∈ V)
8	6, 7	mprg 3070	. . . . 5 ⊢ dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ω
9		peano1 7825	. . . . . 6 ⊢ ∅ ∈ ω
10	9	ne0ii 4297	. . . . 5 ⊢ ω ≠ ∅
11	8, 10	eqnetri 3014	. . . 4 ⊢ dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅
12		dm0rn0 5880	. . . . 5 ⊢ (dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = ∅)
13	12	necon3bii 2996	. . . 4 ⊢ (dom (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅ ↔ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅)
14	11, 13	mpbi 229	. . 3 ⊢ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)) ≠ ∅
15		fri 5593	. . 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 6855	. . . . . . 7 ⊢ (𝐹‘𝑤) ∈ V
18		eqid 2736	. . . . . . 7 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) = (𝑤 ∈ ω ↦ (𝐹‘𝑤))
19	17, 18	fnmpti 6644	. . . . . 6 ⊢ (𝑤 ∈ ω ↦ (𝐹‘𝑤)) Fn ω
20		fvelrnb 6903	. . . . . 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 7827	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → suc 𝑥 ∈ ω)
23		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑤 = suc 𝑥 → (𝐹‘𝑤) = (𝐹‘suc 𝑥))
24		fvex 6855	. . . . . . . . . . 11 ⊢ (𝐹‘suc 𝑥) ∈ V
25	23, 18, 24	fvmpt 6948	. . . . . . . . . 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 7031	. . . . . . . . . 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 2839	. . . . . . . 8 ⊢ (𝑥 ∈ ω → (𝐹‘suc 𝑥) ∈ ran (𝑤 ∈ ω ↦ (𝐹‘𝑤)))
30		epel 5540	. . . . . . . . . . . 12 ⊢ (𝑧 E 𝑦 ↔ 𝑧 ∈ 𝑦)
31		eleq1 2825	. . . . . . . . . . . 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 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)
35	33, 34	bitr4di 288	. . . . . . . . 9 ⊢ (𝑧 = (𝐹‘suc 𝑥) → (¬ 𝑧 E 𝑦 ↔ (𝐹‘suc 𝑥) ∉ 𝑦))
36	35	rspccv 3578	. . . . . . . 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 6842	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝐹‘𝑤) = (𝐹‘𝑥))
39		fvex 6855	. . . . . . . . . . 11 ⊢ (𝐹‘𝑥) ∈ V
40	38, 18, 39	fvmpt 6948	. . . . . . . . . 10 ⊢ (𝑥 ∈ ω → ((𝑤 ∈ ω ↦ (𝐹‘𝑤))‘𝑥) = (𝐹‘𝑥))
41		eqeq1 2740	. . . . . . . . . 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 3055	. . . . . . . . . 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 3162	. . . . 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 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 3145	. 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 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)