Theorem mirimanoffr 36627

Description: Relative version of mirimanoff 36628 (which corresponds to the case 𝑉 = V).

Note that the definition of 𝑀 presupposes that ω is available as a set (omex 9671), which is equivalent to assuming ax-inf2 9669. (Contributed by Adrian Ducourtial, 13-Oct-2026.)

Hypothesis

Ref	Expression
mirimanoffr.1	⊢ 𝑀 = {𝑥 ∈ 𝑉 ∣ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))}

Assertion

Ref	Expression
mirimanoffr	⊢ 𝑀 ∉ 𝑉

Distinct variable groups:   𝑓,𝑛,𝑥,𝑀   𝑓,𝑉,𝑛,𝑥

Proof of Theorem mirimanoffr

Dummy variables 𝑚 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omex 9671	. . . . . . . . 9 ⊢ ω ∈ V
2	1	mptex 7252	. . . . . . . 8 ⊢ (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) ∈ V
3	2	a1i 11	. . . . . . 7 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) → (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) ∈ V)
4		fvex 6921	. . . . . . . . . 10 ⊢ (𝑓‘suc 𝑚) ∈ V
5		eqid 2763	. . . . . . . . . 10 ⊢ (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) = (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))
6	4, 5	fnmpti 6704	. . . . . . . . 9 ⊢ (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) Fn ω
7	6	a1i 11	. . . . . . . 8 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) → (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) Fn ω)
8		peano1 7924	. . . . . . . . . 10 ⊢ ∅ ∈ ω
9		suceq 6449	. . . . . . . . . . . 12 ⊢ (𝑚 = ∅ → suc 𝑚 = suc ∅)
10	9	fveq2d 6912	. . . . . . . . . . 11 ⊢ (𝑚 = ∅ → (𝑓‘suc 𝑚) = (𝑓‘suc ∅))
11		fvex 6921	. . . . . . . . . . 11 ⊢ (𝑓‘suc ∅) ∈ V
12	10, 5, 11	fvmpt 7016	. . . . . . . . . 10 ⊢ (∅ ∈ ω → ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘∅) = (𝑓‘suc ∅))
13	8, 12	ax-mp 5	. . . . . . . . 9 ⊢ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘∅) = (𝑓‘suc ∅)
14	13	a1i 11	. . . . . . . 8 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) → ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘∅) = (𝑓‘suc ∅))
15		suceq 6449	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑚 → suc 𝑛 = suc 𝑚)
16	15	fveq2d 6912	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝑓‘suc 𝑛) = (𝑓‘suc 𝑚))
17		fveq2 6908	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑚 → (𝑓‘𝑛) = (𝑓‘𝑚))
18	16, 17	eleq12d 2858	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → ((𝑓‘suc 𝑛) ∈ (𝑓‘𝑛) ↔ (𝑓‘suc 𝑚) ∈ (𝑓‘𝑚)))
19	18	cbvralvw 3244	. . . . . . . . 9 ⊢ (∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛) ↔ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑓‘𝑚))
20		suceq 6449	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 = suc 𝑛 → suc 𝑚 = suc suc 𝑛)
21	20	fveq2d 6912	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = suc 𝑛 → (𝑓‘suc 𝑚) = (𝑓‘suc suc 𝑛))
22		fveq2 6908	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = suc 𝑛 → (𝑓‘𝑚) = (𝑓‘suc 𝑛))
23	21, 22	eleq12d 2858	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = suc 𝑛 → ((𝑓‘suc 𝑚) ∈ (𝑓‘𝑚) ↔ (𝑓‘suc suc 𝑛) ∈ (𝑓‘suc 𝑛)))
24	23	rspcv 3576	. . . . . . . . . . . . . 14 ⊢ (suc 𝑛 ∈ ω → (∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑓‘𝑚) → (𝑓‘suc suc 𝑛) ∈ (𝑓‘suc 𝑛)))
25		peano2 7925	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ω → suc 𝑛 ∈ ω)
26	24, 25	syl11 34	. . . . . . . . . . . . 13 ⊢ (∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑓‘𝑚) → (𝑛 ∈ ω → (𝑓‘suc suc 𝑛) ∈ (𝑓‘suc 𝑛)))
27	26	3ad2ant3 1153	. . . . . . . . . . . 12 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑓‘𝑚)) → (𝑛 ∈ ω → (𝑓‘suc suc 𝑛) ∈ (𝑓‘suc 𝑛)))
28	27	imp 412	. . . . . . . . . . 11 ⊢ (((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑓‘𝑚)) ∧ 𝑛 ∈ ω) → (𝑓‘suc suc 𝑛) ∈ (𝑓‘suc 𝑛))
29		eqidd 2764	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ω → (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) = (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)))
30		fvexd 6923	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ω → (𝑓‘suc suc 𝑛) ∈ V)
31	21, 29, 25, 30	fvmptd4 7041	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘suc 𝑛) = (𝑓‘suc suc 𝑛))
32		suceq 6449	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑛 → suc 𝑚 = suc 𝑛)
33	32	fveq2d 6912	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑛 → (𝑓‘suc 𝑚) = (𝑓‘suc 𝑛))
34		fvex 6921	. . . . . . . . . . . . . 14 ⊢ (𝑓‘suc 𝑛) ∈ V
35	33, 5, 34	fvmpt 7016	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ω → ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘𝑛) = (𝑓‘suc 𝑛))
36	31, 35	eleq12d 2858	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ω → (((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘𝑛) ↔ (𝑓‘suc suc 𝑛) ∈ (𝑓‘suc 𝑛)))
37	36	adantl 487	. . . . . . . . . . 11 ⊢ (((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑓‘𝑚)) ∧ 𝑛 ∈ ω) → (((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘𝑛) ↔ (𝑓‘suc suc 𝑛) ∈ (𝑓‘suc 𝑛)))
38	28, 37	mpbird 260	. . . . . . . . . 10 ⊢ (((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑓‘𝑚)) ∧ 𝑛 ∈ ω) → ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘𝑛))
39	38	ralrimiva 3158	. . . . . . . . 9 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑚 ∈ ω (𝑓‘suc 𝑚) ∈ (𝑓‘𝑚)) → ∀𝑛 ∈ ω ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘𝑛))
40	19, 39	syl3an3b 1433	. . . . . . . 8 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) → ∀𝑛 ∈ ω ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘𝑛))
41	7, 14, 40	3jca 1146	. . . . . . 7 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) → ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) Fn ω ∧ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘∅) = (𝑓‘suc ∅) ∧ ∀𝑛 ∈ ω ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘𝑛)))
42		fneq1 6652	. . . . . . . 8 ⊢ (𝑔 = (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) → (𝑔 Fn ω ↔ (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) Fn ω))
43		fveq1 6907	. . . . . . . . 9 ⊢ (𝑔 = (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) → (𝑔‘∅) = ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘∅))
44	43	eqeq1d 2765	. . . . . . . 8 ⊢ (𝑔 = (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) → ((𝑔‘∅) = (𝑓‘suc ∅) ↔ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘∅) = (𝑓‘suc ∅)))
45		fveq1 6907	. . . . . . . . . 10 ⊢ (𝑔 = (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) → (𝑔‘suc 𝑛) = ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘suc 𝑛))
46		fveq1 6907	. . . . . . . . . 10 ⊢ (𝑔 = (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) → (𝑔‘𝑛) = ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘𝑛))
47	45, 46	eleq12d 2858	. . . . . . . . 9 ⊢ (𝑔 = (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) → ((𝑔‘suc 𝑛) ∈ (𝑔‘𝑛) ↔ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘𝑛)))
48	47	ralbidv 3189	. . . . . . . 8 ⊢ (𝑔 = (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) → (∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛) ↔ ∀𝑛 ∈ ω ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘𝑛)))
49	42, 44, 48	3anbi123d 1465	. . . . . . 7 ⊢ (𝑔 = (𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) → ((𝑔 Fn ω ∧ (𝑔‘∅) = (𝑓‘suc ∅) ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛)) ↔ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚)) Fn ω ∧ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘∅) = (𝑓‘suc ∅) ∧ ∀𝑛 ∈ ω ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ (𝑓‘suc 𝑚))‘𝑛))))
50	3, 41, 49	spcedv 3556	. . . . . 6 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) → ∃𝑔(𝑔 Fn ω ∧ (𝑔‘∅) = (𝑓‘suc ∅) ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛)))
51		suceq 6449	. . . . . . . . . . . . . 14 ⊢ (𝑛 = ∅ → suc 𝑛 = suc ∅)
52	51	fveq2d 6912	. . . . . . . . . . . . 13 ⊢ (𝑛 = ∅ → (𝑓‘suc 𝑛) = (𝑓‘suc ∅))
53		fveq2 6908	. . . . . . . . . . . . 13 ⊢ (𝑛 = ∅ → (𝑓‘𝑛) = (𝑓‘∅))
54	52, 53	eleq12d 2858	. . . . . . . . . . . 12 ⊢ (𝑛 = ∅ → ((𝑓‘suc 𝑛) ∈ (𝑓‘𝑛) ↔ (𝑓‘suc ∅) ∈ (𝑓‘∅)))
55	54	rspcv 3576	. . . . . . . . . . 11 ⊢ (∅ ∈ ω → (∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛) → (𝑓‘suc ∅) ∈ (𝑓‘∅)))
56	8, 55	ax-mp 5	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛) → (𝑓‘suc ∅) ∈ (𝑓‘∅))
57	56	3ad2ant3 1153	. . . . . . . . 9 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) → (𝑓‘suc ∅) ∈ (𝑓‘∅))
58		eleq2 2853	. . . . . . . . . 10 ⊢ ((𝑓‘∅) = 𝑀 → ((𝑓‘suc ∅) ∈ (𝑓‘∅) ↔ (𝑓‘suc ∅) ∈ 𝑀))
59	58	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) → ((𝑓‘suc ∅) ∈ (𝑓‘∅) ↔ (𝑓‘suc ∅) ∈ 𝑀))
60	57, 59	mpbid 235	. . . . . . . 8 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) → (𝑓‘suc ∅) ∈ 𝑀)
61		eqeq2 2775	. . . . . . . . . . . 12 ⊢ (𝑥 = (𝑓‘suc ∅) → ((𝑔‘∅) = 𝑥 ↔ (𝑔‘∅) = (𝑓‘suc ∅)))
62	61	3anbi2d 1470	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑓‘suc ∅) → ((𝑔 Fn ω ∧ (𝑔‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛)) ↔ (𝑔 Fn ω ∧ (𝑔‘∅) = (𝑓‘suc ∅) ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛))))
63	62	exbidv 1956	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘suc ∅) → (∃𝑔(𝑔 Fn ω ∧ (𝑔‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛)) ↔ ∃𝑔(𝑔 Fn ω ∧ (𝑔‘∅) = (𝑓‘suc ∅) ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛))))
64	63	notbid 321	. . . . . . . . 9 ⊢ (𝑥 = (𝑓‘suc ∅) → (¬ ∃𝑔(𝑔 Fn ω ∧ (𝑔‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛)) ↔ ¬ ∃𝑔(𝑔 Fn ω ∧ (𝑔‘∅) = (𝑓‘suc ∅) ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛))))
65		mirimanoffr.1	. . . . . . . . . 10 ⊢ 𝑀 = {𝑥 ∈ 𝑉 ∣ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))}
66		fneq1 6652	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (𝑓 Fn ω ↔ 𝑔 Fn ω))
67		fveq1 6907	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅))
68	67	eqeq1d 2765	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → ((𝑓‘∅) = 𝑥 ↔ (𝑔‘∅) = 𝑥))
69		fveq1 6907	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝑓‘suc 𝑛) = (𝑔‘suc 𝑛))
70		fveq1 6907	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑛) = (𝑔‘𝑛))
71	69, 70	eleq12d 2858	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → ((𝑓‘suc 𝑛) ∈ (𝑓‘𝑛) ↔ (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛)))
72	71	ralbidv 3189	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → (∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛) ↔ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛)))
73	66, 68, 72	3anbi123d 1465	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) ↔ (𝑔 Fn ω ∧ (𝑔‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛))))
74	73	cbvexvw 2072	. . . . . . . . . . 11 ⊢ (∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) ↔ ∃𝑔(𝑔 Fn ω ∧ (𝑔‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛)))
75	74	notbii 323	. . . . . . . . . 10 ⊢ (¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) ↔ ¬ ∃𝑔(𝑔 Fn ω ∧ (𝑔‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛)))
76	65, 75	rabbieq 3424	. . . . . . . . 9 ⊢ 𝑀 = {𝑥 ∈ 𝑉 ∣ ¬ ∃𝑔(𝑔 Fn ω ∧ (𝑔‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛))}
77	64, 76	elrab2 3652	. . . . . . . 8 ⊢ ((𝑓‘suc ∅) ∈ 𝑀 ↔ ((𝑓‘suc ∅) ∈ 𝑉 ∧ ¬ ∃𝑔(𝑔 Fn ω ∧ (𝑔‘∅) = (𝑓‘suc ∅) ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛))))
78	60, 77	sylib 221	. . . . . . 7 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) → ((𝑓‘suc ∅) ∈ 𝑉 ∧ ¬ ∃𝑔(𝑔 Fn ω ∧ (𝑔‘∅) = (𝑓‘suc ∅) ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛))))
79	78	simprd 501	. . . . . 6 ⊢ ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) → ¬ ∃𝑔(𝑔 Fn ω ∧ (𝑔‘∅) = (𝑓‘suc ∅) ∧ ∀𝑛 ∈ ω (𝑔‘suc 𝑛) ∈ (𝑔‘𝑛)))
80	50, 79	pm2.65i 196	. . . . 5 ⊢ ¬ (𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))
81	80	a1i 11	. . . 4 ⊢ (𝑀 ∈ 𝑉 → ¬ (𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)))
82	81	nexdv 1971	. . 3 ⊢ (𝑀 ∈ 𝑉 → ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)))
83		pm2.24 125	. . . 4 ⊢ (𝑀 ∈ 𝑉 → (¬ 𝑀 ∈ 𝑉 → ¬ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))))
84		idd 25	. . . 4 ⊢ (𝑀 ∈ 𝑉 → (¬ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) → ¬ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))))
85	1	mptex 7252	. . . . . . . . 9 ⊢ (𝑚 ∈ ω ↦ 𝑀) ∈ V
86	85	isseti 3472	. . . . . . . 8 ⊢ ∃𝑓 𝑓 = (𝑚 ∈ ω ↦ 𝑀)
87		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑀) → 𝑀 ∈ 𝑉)
88	87	ralrimivw 3162	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑀) → ∀𝑚 ∈ ω 𝑀 ∈ 𝑉)
89		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑚 ∈ ω ↦ 𝑀) = (𝑚 ∈ ω ↦ 𝑀)
90	89	fnmpt 6701	. . . . . . . . . . . . 13 ⊢ (∀𝑚 ∈ ω 𝑀 ∈ 𝑉 → (𝑚 ∈ ω ↦ 𝑀) Fn ω)
91	88, 90	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑀) → (𝑚 ∈ ω ↦ 𝑀) Fn ω)
92		eqidd 2764	. . . . . . . . . . . . . 14 ⊢ (𝑚 = ∅ → 𝑀 = 𝑀)
93	92, 89	fvmptg 7014	. . . . . . . . . . . . 13 ⊢ ((∅ ∈ ω ∧ 𝑀 ∈ 𝑉) → ((𝑚 ∈ ω ↦ 𝑀)‘∅) = 𝑀)
94	8, 87, 93	sylancr 599	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑀) → ((𝑚 ∈ ω ↦ 𝑀)‘∅) = 𝑀)
95		eqidd 2764	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑚 = suc 𝑛 → 𝑀 = 𝑀)
96	95, 89	fvmptg 7014	. . . . . . . . . . . . . . . . . 18 ⊢ ((suc 𝑛 ∈ ω ∧ 𝑀 ∈ 𝑉) → ((𝑚 ∈ ω ↦ 𝑀)‘suc 𝑛) = 𝑀)
97	25, 96	sylan 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ω ∧ 𝑀 ∈ 𝑉) → ((𝑚 ∈ ω ↦ 𝑀)‘suc 𝑛) = 𝑀)
98		eqidd 2764	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 = 𝑛 → 𝑀 = 𝑀)
99	98, 89	fvmptg 7014	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑛 ∈ ω ∧ 𝑀 ∈ 𝑉) → ((𝑚 ∈ ω ↦ 𝑀)‘𝑛) = 𝑀)
100	97, 99	eleq12d 2858	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛 ∈ ω ∧ 𝑀 ∈ 𝑉) → (((𝑚 ∈ ω ↦ 𝑀)‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ 𝑀)‘𝑛) ↔ 𝑀 ∈ 𝑀))
101	100	biimprd 251	. . . . . . . . . . . . . . 15 ⊢ ((𝑛 ∈ ω ∧ 𝑀 ∈ 𝑉) → (𝑀 ∈ 𝑀 → ((𝑚 ∈ ω ↦ 𝑀)‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ 𝑀)‘𝑛)))
102	101	ancoms 464	. . . . . . . . . . . . . 14 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑛 ∈ ω) → (𝑀 ∈ 𝑀 → ((𝑚 ∈ ω ↦ 𝑀)‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ 𝑀)‘𝑛)))
103	102	impancom 457	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑀) → (𝑛 ∈ ω → ((𝑚 ∈ ω ↦ 𝑀)‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ 𝑀)‘𝑛)))
104	103	ralrimiv 3157	. . . . . . . . . . . 12 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑀) → ∀𝑛 ∈ ω ((𝑚 ∈ ω ↦ 𝑀)‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ 𝑀)‘𝑛))
105	91, 94, 104	3jca 1146	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑀) → ((𝑚 ∈ ω ↦ 𝑀) Fn ω ∧ ((𝑚 ∈ ω ↦ 𝑀)‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω ((𝑚 ∈ ω ↦ 𝑀)‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ 𝑀)‘𝑛)))
106		fneq1 6652	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑚 ∈ ω ↦ 𝑀) → (𝑓 Fn ω ↔ (𝑚 ∈ ω ↦ 𝑀) Fn ω))
107		fveq1 6907	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑚 ∈ ω ↦ 𝑀) → (𝑓‘∅) = ((𝑚 ∈ ω ↦ 𝑀)‘∅))
108	107	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑚 ∈ ω ↦ 𝑀) → ((𝑓‘∅) = 𝑀 ↔ ((𝑚 ∈ ω ↦ 𝑀)‘∅) = 𝑀))
109		fveq1 6907	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑚 ∈ ω ↦ 𝑀) → (𝑓‘suc 𝑛) = ((𝑚 ∈ ω ↦ 𝑀)‘suc 𝑛))
110		fveq1 6907	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑚 ∈ ω ↦ 𝑀) → (𝑓‘𝑛) = ((𝑚 ∈ ω ↦ 𝑀)‘𝑛))
111	109, 110	eleq12d 2858	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑚 ∈ ω ↦ 𝑀) → ((𝑓‘suc 𝑛) ∈ (𝑓‘𝑛) ↔ ((𝑚 ∈ ω ↦ 𝑀)‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ 𝑀)‘𝑛)))
112	111	ralbidv 3189	. . . . . . . . . . . 12 ⊢ (𝑓 = (𝑚 ∈ ω ↦ 𝑀) → (∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛) ↔ ∀𝑛 ∈ ω ((𝑚 ∈ ω ↦ 𝑀)‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ 𝑀)‘𝑛)))
113	106, 108, 112	3anbi123d 1465	. . . . . . . . . . 11 ⊢ (𝑓 = (𝑚 ∈ ω ↦ 𝑀) → ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) ↔ ((𝑚 ∈ ω ↦ 𝑀) Fn ω ∧ ((𝑚 ∈ ω ↦ 𝑀)‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω ((𝑚 ∈ ω ↦ 𝑀)‘suc 𝑛) ∈ ((𝑚 ∈ ω ↦ 𝑀)‘𝑛))))
114	105, 113	syl5ibrcom 250	. . . . . . . . . 10 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑀) → (𝑓 = (𝑚 ∈ ω ↦ 𝑀) → (𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))))
115	114	eximdv 1952	. . . . . . . . 9 ⊢ ((𝑀 ∈ 𝑉 ∧ 𝑀 ∈ 𝑀) → (∃𝑓 𝑓 = (𝑚 ∈ ω ↦ 𝑀) → ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))))
116	115	ex 418	. . . . . . . 8 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ 𝑀 → (∃𝑓 𝑓 = (𝑚 ∈ ω ↦ 𝑀) → ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)))))
117	86, 116	mpii 47	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → (𝑀 ∈ 𝑀 → ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))))
118	82, 117	mtod 201	. . . . . 6 ⊢ (𝑀 ∈ 𝑉 → ¬ 𝑀 ∈ 𝑀)
119		eqeq2 2775	. . . . . . . . . 10 ⊢ (𝑥 = 𝑀 → ((𝑓‘∅) = 𝑥 ↔ (𝑓‘∅) = 𝑀))
120	119	3anbi2d 1470	. . . . . . . . 9 ⊢ (𝑥 = 𝑀 → ((𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) ↔ (𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))))
121	120	exbidv 1956	. . . . . . . 8 ⊢ (𝑥 = 𝑀 → (∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) ↔ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))))
122	121	notbid 321	. . . . . . 7 ⊢ (𝑥 = 𝑀 → (¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑥 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)) ↔ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))))
123	122, 65	elrab2 3652	. . . . . 6 ⊢ (𝑀 ∈ 𝑀 ↔ (𝑀 ∈ 𝑉 ∧ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))))
124	118, 123	sylnib 331	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → ¬ (𝑀 ∈ 𝑉 ∧ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))))
125		ianor 997	. . . . 5 ⊢ (¬ (𝑀 ∈ 𝑉 ∧ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))) ↔ (¬ 𝑀 ∈ 𝑉 ∨ ¬ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))))
126	124, 125	sylib 221	. . . 4 ⊢ (𝑀 ∈ 𝑉 → (¬ 𝑀 ∈ 𝑉 ∨ ¬ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛))))
127	83, 84, 126	mpjaod 874	. . 3 ⊢ (𝑀 ∈ 𝑉 → ¬ ¬ ∃𝑓(𝑓 Fn ω ∧ (𝑓‘∅) = 𝑀 ∧ ∀𝑛 ∈ ω (𝑓‘suc 𝑛) ∈ (𝑓‘𝑛)))
128	82, 127	pm2.65i 196	. 2 ⊢ ¬ 𝑀 ∈ 𝑉
129	128	nelir 3068	1 ⊢ 𝑀 ∉ 𝑉

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ∉ wnel 3065  ∀wral 3080  {crab 3416  Vcvv 3454  ∅c0 4281   ↦ cmpt 5189  suc csuc 6382   Fn wfn 6553  ‘cfv 6558  ωcom 7901

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7775  ax-inf2 9669

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-om 7902

This theorem is used by:  mirimanoff  36628