Theorem ennnfonelemhom 13284

Description: Lemma for ennnfone 13294. The sequences in 𝐻 increase in length without bound if you go out far enough. (Contributed by Jim Kingdon, 19-Jul-2023.)

Hypotheses

Ref	Expression
ennnfonelemh.dceq	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
ennnfonelemh.f	⊢ (𝜑 → 𝐹:ω–onto→𝐴)
ennnfonelemh.ne	⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
ennnfonelemh.g	⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
ennnfonelemh.n	⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
ennnfonelemh.j	⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
ennnfonelemh.h	⊢ 𝐻 = seq0(𝐺, 𝐽)
ennnfonelemhom.m	⊢ (𝜑 → 𝑀 ∈ ω)

Assertion

Ref	Expression
ennnfonelemhom	⊢ (𝜑 → ∃𝑖 ∈ ℕ0 𝑀 ∈ dom (𝐻‘𝑖))

Distinct variable groups:   𝑖,𝐻,𝑘,𝑗,𝑥,𝑦   𝑖,𝑀   𝜑,𝑖,𝑘,𝑥,𝑦,𝑗   𝜑,𝑛   𝑥,𝑁,𝑦,𝑗,𝑘   𝑛,𝑁   𝑗,𝐺   𝑘,𝐹,𝑥,𝑦,𝑗   𝑛,𝐹,𝑗   𝑥,𝐴,𝑦,𝑗   𝑗,𝐽   𝑥,𝑖,𝑦,𝑗   𝑖,𝑛,𝐻,𝑘

Allowed substitution hints:   𝐴(𝑖,𝑘,𝑛)   𝐹(𝑖)   𝐺(𝑥,𝑦,𝑖,𝑘,𝑛)   𝐽(𝑥,𝑦,𝑖,𝑘,𝑛)   𝑀(𝑥,𝑦,𝑗,𝑘,𝑛)   𝑁(𝑖)

Proof of Theorem ennnfonelemhom

Dummy variables 𝑞 𝑤 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ennnfonelemhom.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ω)
2		eleq1 2301	. . . . 5 ⊢ (𝑤 = ∅ → (𝑤 ∈ dom (𝐻‘𝑖) ↔ ∅ ∈ dom (𝐻‘𝑖)))
3	2	rexbidv 2551	. . . 4 ⊢ (𝑤 = ∅ → (∃𝑖 ∈ ℕ0 𝑤 ∈ dom (𝐻‘𝑖) ↔ ∃𝑖 ∈ ℕ0 ∅ ∈ dom (𝐻‘𝑖)))
4	3	imbi2d 230	. . 3 ⊢ (𝑤 = ∅ → ((𝜑 → ∃𝑖 ∈ ℕ0 𝑤 ∈ dom (𝐻‘𝑖)) ↔ (𝜑 → ∃𝑖 ∈ ℕ0 ∅ ∈ dom (𝐻‘𝑖))))
5		eleq1 2301	. . . . 5 ⊢ (𝑤 = 𝑘 → (𝑤 ∈ dom (𝐻‘𝑖) ↔ 𝑘 ∈ dom (𝐻‘𝑖)))
6	5	rexbidv 2551	. . . 4 ⊢ (𝑤 = 𝑘 → (∃𝑖 ∈ ℕ0 𝑤 ∈ dom (𝐻‘𝑖) ↔ ∃𝑖 ∈ ℕ0 𝑘 ∈ dom (𝐻‘𝑖)))
7	6	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑘 → ((𝜑 → ∃𝑖 ∈ ℕ0 𝑤 ∈ dom (𝐻‘𝑖)) ↔ (𝜑 → ∃𝑖 ∈ ℕ0 𝑘 ∈ dom (𝐻‘𝑖))))
8		eleq1 2301	. . . . 5 ⊢ (𝑤 = suc 𝑘 → (𝑤 ∈ dom (𝐻‘𝑖) ↔ suc 𝑘 ∈ dom (𝐻‘𝑖)))
9	8	rexbidv 2551	. . . 4 ⊢ (𝑤 = suc 𝑘 → (∃𝑖 ∈ ℕ0 𝑤 ∈ dom (𝐻‘𝑖) ↔ ∃𝑖 ∈ ℕ0 suc 𝑘 ∈ dom (𝐻‘𝑖)))
10	9	imbi2d 230	. . 3 ⊢ (𝑤 = suc 𝑘 → ((𝜑 → ∃𝑖 ∈ ℕ0 𝑤 ∈ dom (𝐻‘𝑖)) ↔ (𝜑 → ∃𝑖 ∈ ℕ0 suc 𝑘 ∈ dom (𝐻‘𝑖))))
11		eleq1 2301	. . . . 5 ⊢ (𝑤 = 𝑀 → (𝑤 ∈ dom (𝐻‘𝑖) ↔ 𝑀 ∈ dom (𝐻‘𝑖)))
12	11	rexbidv 2551	. . . 4 ⊢ (𝑤 = 𝑀 → (∃𝑖 ∈ ℕ0 𝑤 ∈ dom (𝐻‘𝑖) ↔ ∃𝑖 ∈ ℕ0 𝑀 ∈ dom (𝐻‘𝑖)))
13	12	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑀 → ((𝜑 → ∃𝑖 ∈ ℕ0 𝑤 ∈ dom (𝐻‘𝑖)) ↔ (𝜑 → ∃𝑖 ∈ ℕ0 𝑀 ∈ dom (𝐻‘𝑖))))
14		1nn0 9558	. . . 4 ⊢ 1 ∈ ℕ0
15		0ex 4255	. . . . . 6 ⊢ ∅ ∈ V
16	15	snid 3736	. . . . 5 ⊢ ∅ ∈ {∅}
17		ennnfonelemh.dceq	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
18		ennnfonelemh.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:ω–onto→𝐴)
19		ennnfonelemh.ne	. . . . . . . 8 ⊢ (𝜑 → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
20		ennnfonelemh.g	. . . . . . . 8 ⊢ 𝐺 = (𝑥 ∈ (𝐴 ↑pm ω), 𝑦 ∈ ω ↦ if((𝐹‘𝑦) ∈ (𝐹 “ 𝑦), 𝑥, (𝑥 ∪ {⟨dom 𝑥, (𝐹‘𝑦)⟩})))
21		ennnfonelemh.n	. . . . . . . 8 ⊢ 𝑁 = frec((𝑥 ∈ ℤ ↦ (𝑥 + 1)), 0)
22		ennnfonelemh.j	. . . . . . . 8 ⊢ 𝐽 = (𝑥 ∈ ℕ0 ↦ if(𝑥 = 0, ∅, (◡𝑁‘(𝑥 − 1))))
23		ennnfonelemh.h	. . . . . . . 8 ⊢ 𝐻 = seq0(𝐺, 𝐽)
24	17, 18, 19, 20, 21, 22, 23	ennnfonelem1 13276	. . . . . . 7 ⊢ (𝜑 → (𝐻‘1) = {⟨∅, (𝐹‘∅)⟩})
25	24	dmeqd 4978	. . . . . 6 ⊢ (𝜑 → dom (𝐻‘1) = dom {⟨∅, (𝐹‘∅)⟩})
26		peano1 4736	. . . . . . . 8 ⊢ ∅ ∈ ω
27		fof 5610	. . . . . . . . . 10 ⊢ (𝐹:ω–onto→𝐴 → 𝐹:ω⟶𝐴)
28	18, 27	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:ω⟶𝐴)
29	26	a1i 9	. . . . . . . . 9 ⊢ (𝜑 → ∅ ∈ ω)
30	28, 29	ffvelcdmd 5835	. . . . . . . 8 ⊢ (𝜑 → (𝐹‘∅) ∈ 𝐴)
31		fnsng 5423	. . . . . . . 8 ⊢ ((∅ ∈ ω ∧ (𝐹‘∅) ∈ 𝐴) → {⟨∅, (𝐹‘∅)⟩} Fn {∅})
32	26, 30, 31	sylancr 418	. . . . . . 7 ⊢ (𝜑 → {⟨∅, (𝐹‘∅)⟩} Fn {∅})
33		fndm 5475	. . . . . . 7 ⊢ ({⟨∅, (𝐹‘∅)⟩} Fn {∅} → dom {⟨∅, (𝐹‘∅)⟩} = {∅})
34	32, 33	syl 14	. . . . . 6 ⊢ (𝜑 → dom {⟨∅, (𝐹‘∅)⟩} = {∅})
35	25, 34	eqtrd 2271	. . . . 5 ⊢ (𝜑 → dom (𝐻‘1) = {∅})
36	16, 35	eleqtrrid 2328	. . . 4 ⊢ (𝜑 → ∅ ∈ dom (𝐻‘1))
37		fveq2 5690	. . . . . . 7 ⊢ (𝑖 = 1 → (𝐻‘𝑖) = (𝐻‘1))
38	37	dmeqd 4978	. . . . . 6 ⊢ (𝑖 = 1 → dom (𝐻‘𝑖) = dom (𝐻‘1))
39	38	eleq2d 2308	. . . . 5 ⊢ (𝑖 = 1 → (∅ ∈ dom (𝐻‘𝑖) ↔ ∅ ∈ dom (𝐻‘1)))
40	39	rspcev 2929	. . . 4 ⊢ ((1 ∈ ℕ0 ∧ ∅ ∈ dom (𝐻‘1)) → ∃𝑖 ∈ ℕ0 ∅ ∈ dom (𝐻‘𝑖))
41	14, 36, 40	sylancr 418	. . 3 ⊢ (𝜑 → ∃𝑖 ∈ ℕ0 ∅ ∈ dom (𝐻‘𝑖))
42	17	ad3antrrr 496	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
43	18	ad3antrrr 496	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) → 𝐹:ω–onto→𝐴)
44	19	ad3antrrr 496	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
45		fveq2 5690	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑎 → (𝐹‘𝑘) = (𝐹‘𝑎))
46	45	neeq1d 2438	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑎 → ((𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ (𝐹‘𝑎) ≠ (𝐹‘𝑗)))
47	46	ralbidv 2550	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑎 → (∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑗 ∈ suc 𝑛(𝐹‘𝑎) ≠ (𝐹‘𝑗)))
48	47	cbvrexv 2787	. . . . . . . . . . 11 ⊢ (∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∃𝑎 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑎) ≠ (𝐹‘𝑗))
49	48	ralbii 2556	. . . . . . . . . 10 ⊢ (∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗) ↔ ∀𝑛 ∈ ω ∃𝑎 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑎) ≠ (𝐹‘𝑗))
50	44, 49	sylib 122	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) → ∀𝑛 ∈ ω ∃𝑎 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑎) ≠ (𝐹‘𝑗))
51		simplr 533	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) → 𝑖 ∈ ℕ0)
52	42, 43, 50, 20, 21, 22, 23, 51	ennnfonelemex 13283	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) → ∃𝑞 ∈ ℕ0 dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞))
53	42	ad2antrr 492	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) ∧ 𝑞 ∈ ℕ0) ∧ dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦)
54	43	ad2antrr 492	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) ∧ 𝑞 ∈ ℕ0) ∧ dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞)) → 𝐹:ω–onto→𝐴)
55	44	ad2antrr 492	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) ∧ 𝑞 ∈ ℕ0) ∧ dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞)) → ∀𝑛 ∈ ω ∃𝑘 ∈ ω ∀𝑗 ∈ suc 𝑛(𝐹‘𝑘) ≠ (𝐹‘𝑗))
56		simplr 533	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) ∧ 𝑞 ∈ ℕ0) ∧ dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞)) → 𝑞 ∈ ℕ0)
57	53, 54, 55, 20, 21, 22, 23, 56	ennnfonelemom 13277	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) ∧ 𝑞 ∈ ℕ0) ∧ dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞)) → dom (𝐻‘𝑞) ∈ ω)
58		nnord 4754	. . . . . . . . . . . . 13 ⊢ (dom (𝐻‘𝑞) ∈ ω → Ord dom (𝐻‘𝑞))
59	57, 58	syl 14	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) ∧ 𝑞 ∈ ℕ0) ∧ dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞)) → Ord dom (𝐻‘𝑞))
60		simpr 110	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) ∧ 𝑞 ∈ ℕ0) ∧ dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞)) → dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞))
61		ordsucss 4646	. . . . . . . . . . . 12 ⊢ (Ord dom (𝐻‘𝑞) → (dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞) → suc dom (𝐻‘𝑖) ⊆ dom (𝐻‘𝑞)))
62	59, 60, 61	sylc 62	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) ∧ 𝑞 ∈ ℕ0) ∧ dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞)) → suc dom (𝐻‘𝑖) ⊆ dom (𝐻‘𝑞))
63		simpr 110	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) → 𝑘 ∈ dom (𝐻‘𝑖))
64	42, 43, 44, 20, 21, 22, 23, 51	ennnfonelemom 13277	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) → dom (𝐻‘𝑖) ∈ ω)
65		nnsucelsuc 6754	. . . . . . . . . . . . . 14 ⊢ (dom (𝐻‘𝑖) ∈ ω → (𝑘 ∈ dom (𝐻‘𝑖) ↔ suc 𝑘 ∈ suc dom (𝐻‘𝑖)))
66	64, 65	syl 14	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) → (𝑘 ∈ dom (𝐻‘𝑖) ↔ suc 𝑘 ∈ suc dom (𝐻‘𝑖)))
67	63, 66	mpbid 147	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) → suc 𝑘 ∈ suc dom (𝐻‘𝑖))
68	67	ad2antrr 492	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) ∧ 𝑞 ∈ ℕ0) ∧ dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞)) → suc 𝑘 ∈ suc dom (𝐻‘𝑖))
69	62, 68	sseldd 3249	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) ∧ 𝑞 ∈ ℕ0) ∧ dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞)) → suc 𝑘 ∈ dom (𝐻‘𝑞))
70	69	ex 115	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) ∧ 𝑞 ∈ ℕ0) → (dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞) → suc 𝑘 ∈ dom (𝐻‘𝑞)))
71	70	reximdva 2652	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) → (∃𝑞 ∈ ℕ0 dom (𝐻‘𝑖) ∈ dom (𝐻‘𝑞) → ∃𝑞 ∈ ℕ0 suc 𝑘 ∈ dom (𝐻‘𝑞)))
72	52, 71	mpd 13	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑘 ∈ ω) ∧ 𝑖 ∈ ℕ0) ∧ 𝑘 ∈ dom (𝐻‘𝑖)) → ∃𝑞 ∈ ℕ0 suc 𝑘 ∈ dom (𝐻‘𝑞))
73	72	rexlimdva2 2671	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → (∃𝑖 ∈ ℕ0 𝑘 ∈ dom (𝐻‘𝑖) → ∃𝑞 ∈ ℕ0 suc 𝑘 ∈ dom (𝐻‘𝑞)))
74		fveq2 5690	. . . . . . . . 9 ⊢ (𝑖 = 𝑞 → (𝐻‘𝑖) = (𝐻‘𝑞))
75	74	dmeqd 4978	. . . . . . . 8 ⊢ (𝑖 = 𝑞 → dom (𝐻‘𝑖) = dom (𝐻‘𝑞))
76	75	eleq2d 2308	. . . . . . 7 ⊢ (𝑖 = 𝑞 → (suc 𝑘 ∈ dom (𝐻‘𝑖) ↔ suc 𝑘 ∈ dom (𝐻‘𝑞)))
77	76	cbvrexv 2787	. . . . . 6 ⊢ (∃𝑖 ∈ ℕ0 suc 𝑘 ∈ dom (𝐻‘𝑖) ↔ ∃𝑞 ∈ ℕ0 suc 𝑘 ∈ dom (𝐻‘𝑞))
78	73, 77	imbitrrdi 162	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ω) → (∃𝑖 ∈ ℕ0 𝑘 ∈ dom (𝐻‘𝑖) → ∃𝑖 ∈ ℕ0 suc 𝑘 ∈ dom (𝐻‘𝑖)))
79	78	expcom 116	. . . 4 ⊢ (𝑘 ∈ ω → (𝜑 → (∃𝑖 ∈ ℕ0 𝑘 ∈ dom (𝐻‘𝑖) → ∃𝑖 ∈ ℕ0 suc 𝑘 ∈ dom (𝐻‘𝑖))))
80	79	a2d 26	. . 3 ⊢ (𝑘 ∈ ω → ((𝜑 → ∃𝑖 ∈ ℕ0 𝑘 ∈ dom (𝐻‘𝑖)) → (𝜑 → ∃𝑖 ∈ ℕ0 suc 𝑘 ∈ dom (𝐻‘𝑖))))
81	4, 7, 10, 13, 41, 80	finds 4742	. 2 ⊢ (𝑀 ∈ ω → (𝜑 → ∃𝑖 ∈ ℕ0 𝑀 ∈ dom (𝐻‘𝑖)))
82	1, 81	mpcom 36	1 ⊢ (𝜑 → ∃𝑖 ∈ ℕ0 𝑀 ∈ dom (𝐻‘𝑖))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   = wceq 1402   ∈ wcel 2209   ≠ wne 2420  ∀wral 2528  ∃wrex 2529   ∪ cun 3218   ⊆ wss 3220  ∅c0 3520  ifcif 3635  {csn 3705  ⟨cop 3708   ↦ cmpt 4187  Ord word 4502  suc csuc 4505  ωcom 4732  ^◡ccnv 4768  dom cdm 4769   “ cima 4772   Fn wfn 5367  ⟶wf 5368  –onto→wfo 5370  ‘cfv 5372  (class class class)co 6075   ∈ cmpo 6077  freccfrec 6651   ↑_pm cpm 6913  0cc0 8169  1c1 8170   + caddc 8172   − cmin 8487  ℕ₀cn0 9542  ℤcz 9623  seqcseq 10862

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-pm 6915  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-seqfrec 10863

This theorem is referenced by:  ennnfonelemdm  13289