Theorem bdayfinbndlem2 28854

Description: Lemma for bdayfinbnd 28855. Conduct the induction. (Contributed by Scott Fenton, 26-Feb-2026.)

Assertion

Ref	Expression
bdayfinbndlem2	⊢ (𝑁 ∈ ℕ0s → ∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))

Distinct variable group:   𝑁,𝑝,𝑥,𝑦,𝑧

Proof of Theorem bdayfinbndlem2

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑞 𝑟 𝑡 𝑤 𝑛 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . . . . . . 8 ⊢ (𝑚 = 0s → (bday‘𝑚) = (bday‘ 0s ))
2		bday0 28197	. . . . . . . 8 ⊢ (bday‘ 0s ) = ∅
3	1, 2	eqtrdi 2811	. . . . . . 7 ⊢ (𝑚 = 0s → (bday‘𝑚) = ∅)
4	3	sseq2d 3940	. . . . . 6 ⊢ (𝑚 = 0s → ((bday‘𝑧) ⊆ (bday‘𝑚) ↔ (bday‘𝑧) ⊆ ∅))
5		ss0b 4350	. . . . . 6 ⊢ ((bday‘𝑧) ⊆ ∅ ↔ (bday‘𝑧) = ∅)
6	4, 5	bitrdi 290	. . . . 5 ⊢ (𝑚 = 0s → ((bday‘𝑧) ⊆ (bday‘𝑚) ↔ (bday‘𝑧) = ∅))
7	6	anbi1d 643	. . . 4 ⊢ (𝑚 = 0s → (((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) ↔ ((bday‘𝑧) = ∅ ∧ 0s ≤s 𝑧)))
8		eqeq2 2772	. . . . 5 ⊢ (𝑚 = 0s → (𝑧 = 𝑚 ↔ 𝑧 = 0s ))
9		breq2 5106	. . . . . . . 8 ⊢ (𝑚 = 0s → ((𝑥 +s 𝑝) <s 𝑚 ↔ (𝑥 +s 𝑝) <s 0s ))
10	9	3anbi3d 1470	. . . . . . 7 ⊢ (𝑚 = 0s → ((𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚) ↔ (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 0s )))
11	10	rexbidv 3186	. . . . . 6 ⊢ (𝑚 = 0s → (∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚) ↔ ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 0s )))
12	11	2rexbidv 3227	. . . . 5 ⊢ (𝑚 = 0s → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 0s )))
13	8, 12	orbi12d 932	. . . 4 ⊢ (𝑚 = 0s → ((𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚)) ↔ (𝑧 = 0s ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 0s ))))
14	7, 13	imbi12d 347	. . 3 ⊢ (𝑚 = 0s → ((((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚))) ↔ (((bday‘𝑧) = ∅ ∧ 0s ≤s 𝑧) → (𝑧 = 0s ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 0s )))))
15	14	ralbidv 3185	. 2 ⊢ (𝑚 = 0s → (∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚))) ↔ ∀𝑧 ∈ No (((bday‘𝑧) = ∅ ∧ 0s ≤s 𝑧) → (𝑧 = 0s ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 0s )))))
16		fveq2 6885	. . . . . 6 ⊢ (𝑚 = 𝑛 → (bday‘𝑚) = (bday‘𝑛))
17	16	sseq2d 3940	. . . . 5 ⊢ (𝑚 = 𝑛 → ((bday‘𝑧) ⊆ (bday‘𝑚) ↔ (bday‘𝑧) ⊆ (bday‘𝑛)))
18	17	anbi1d 643	. . . 4 ⊢ (𝑚 = 𝑛 → (((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) ↔ ((bday‘𝑧) ⊆ (bday‘𝑛) ∧ 0s ≤s 𝑧)))
19		eqeq2 2772	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑧 = 𝑚 ↔ 𝑧 = 𝑛))
20		breq2 5106	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑥 +s 𝑝) <s 𝑚 ↔ (𝑥 +s 𝑝) <s 𝑛))
21	20	3anbi3d 1470	. . . . . . 7 ⊢ (𝑚 = 𝑛 → ((𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚) ↔ (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑛)))
22	21	rexbidv 3186	. . . . . 6 ⊢ (𝑚 = 𝑛 → (∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚) ↔ ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑛)))
23	22	2rexbidv 3227	. . . . 5 ⊢ (𝑚 = 𝑛 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑛)))
24	19, 23	orbi12d 932	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚)) ↔ (𝑧 = 𝑛 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑛))))
25	18, 24	imbi12d 347	. . 3 ⊢ (𝑚 = 𝑛 → ((((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚))) ↔ (((bday‘𝑧) ⊆ (bday‘𝑛) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑛 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑛)))))
26	25	ralbidv 3185	. 2 ⊢ (𝑚 = 𝑛 → (∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚))) ↔ ∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑛) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑛 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑛)))))
27		fveq2 6885	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → (bday‘𝑚) = (bday‘(𝑛 +s 1s )))
28	27	sseq2d 3940	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → ((bday‘𝑧) ⊆ (bday‘𝑚) ↔ (bday‘𝑧) ⊆ (bday‘(𝑛 +s 1s ))))
29	28	anbi1d 643	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → (((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) ↔ ((bday‘𝑧) ⊆ (bday‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑧)))
30		eqeq2 2772	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑧 = 𝑚 ↔ 𝑧 = (𝑛 +s 1s )))
31		breq2 5106	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝑥 +s 𝑝) <s 𝑚 ↔ (𝑥 +s 𝑝) <s (𝑛 +s 1s )))
32	31	3anbi3d 1470	. . . . . . . 8 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚) ↔ (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s ))))
33	32	rexbidv 3186	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚) ↔ ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s ))))
34	33	2rexbidv 3227	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s ))))
35	30, 34	orbi12d 932	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚)) ↔ (𝑧 = (𝑛 +s 1s ) ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s )))))
36	29, 35	imbi12d 347	. . . 4 ⊢ (𝑚 = (𝑛 +s 1s ) → ((((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚))) ↔ (((bday‘𝑧) ⊆ (bday‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑧) → (𝑧 = (𝑛 +s 1s ) ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s ))))))
37	36	ralbidv 3185	. . 3 ⊢ (𝑚 = (𝑛 +s 1s ) → (∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚))) ↔ ∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑧) → (𝑧 = (𝑛 +s 1s ) ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s ))))))
38		fveq2 6885	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (bday‘𝑧) = (bday‘𝑤))
39	38	sseq1d 3939	. . . . . 6 ⊢ (𝑧 = 𝑤 → ((bday‘𝑧) ⊆ (bday‘(𝑛 +s 1s )) ↔ (bday‘𝑤) ⊆ (bday‘(𝑛 +s 1s ))))
40		breq2 5106	. . . . . 6 ⊢ (𝑧 = 𝑤 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑤))
41	39, 40	anbi12d 644	. . . . 5 ⊢ (𝑧 = 𝑤 → (((bday‘𝑧) ⊆ (bday‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑧) ↔ ((bday‘𝑤) ⊆ (bday‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑤)))
42		eqeq1 2764	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑛 +s 1s ) ↔ 𝑤 = (𝑛 +s 1s )))
43		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
44	43	3anbi1d 1468	. . . . . . . . 9 ⊢ (𝑧 = 𝑤 → ((𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s )) ↔ (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s ))))
45	44	rexbidv 3186	. . . . . . . 8 ⊢ (𝑧 = 𝑤 → (∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s )) ↔ ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s ))))
46	45	2rexbidv 3227	. . . . . . 7 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s )) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s ))))
47		oveq1 7427	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑥 +s (𝑦 /su (2s↑s𝑝))) = (𝑎 +s (𝑦 /su (2s↑s𝑝))))
48	47	eqeq2d 2771	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝)))))
49		oveq1 7427	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑥 +s 𝑝) = (𝑎 +s 𝑝))
50	49	breq1d 5112	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ((𝑥 +s 𝑝) <s (𝑛 +s 1s ) ↔ (𝑎 +s 𝑝) <s (𝑛 +s 1s )))
51	48, 50	3anbi13d 1466	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s )) ↔ (𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s (𝑛 +s 1s ))))
52	51	rexbidv 3186	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s )) ↔ ∃𝑝 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s (𝑛 +s 1s ))))
53		oveq1 7427	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → (𝑦 /su (2s↑s𝑝)) = (𝑏 /su (2s↑s𝑝)))
54	53	oveq2d 7436	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝑎 +s (𝑦 /su (2s↑s𝑝))) = (𝑎 +s (𝑏 /su (2s↑s𝑝))))
55	54	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝)))))
56		breq1 5105	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (𝑦 <s (2s↑s𝑝) ↔ 𝑏 <s (2s↑s𝑝)))
57	55, 56	3anbi12d 1465	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → ((𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s (𝑛 +s 1s )) ↔ (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝))) ∧ 𝑏 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s (𝑛 +s 1s ))))
58	57	rexbidv 3186	. . . . . . . . 9 ⊢ (𝑦 = 𝑏 → (∃𝑝 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s (𝑛 +s 1s )) ↔ ∃𝑝 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝))) ∧ 𝑏 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s (𝑛 +s 1s ))))
59		oveq2 7428	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑞 → (2s↑s𝑝) = (2s↑s𝑞))
60	59	oveq2d 7436	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑞 → (𝑏 /su (2s↑s𝑝)) = (𝑏 /su (2s↑s𝑞)))
61	60	oveq2d 7436	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑞 → (𝑎 +s (𝑏 /su (2s↑s𝑝))) = (𝑎 +s (𝑏 /su (2s↑s𝑞))))
62	61	eqeq2d 2771	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑞 → (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞)))))
63	59	breq2d 5114	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑞 → (𝑏 <s (2s↑s𝑝) ↔ 𝑏 <s (2s↑s𝑞)))
64		oveq2 7428	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑞 → (𝑎 +s 𝑝) = (𝑎 +s 𝑞))
65	64	breq1d 5112	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑞 → ((𝑎 +s 𝑝) <s (𝑛 +s 1s ) ↔ (𝑎 +s 𝑞) <s (𝑛 +s 1s )))
66	62, 63, 65	3anbi123d 1464	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → ((𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝))) ∧ 𝑏 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s (𝑛 +s 1s )) ↔ (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑛 +s 1s ))))
67	66	cbvrexvw 3241	. . . . . . . . 9 ⊢ (∃𝑝 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝))) ∧ 𝑏 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s (𝑛 +s 1s )) ↔ ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑛 +s 1s )))
68	58, 67	bitrdi 290	. . . . . . . 8 ⊢ (𝑦 = 𝑏 → (∃𝑝 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑎 +s 𝑝) <s (𝑛 +s 1s )) ↔ ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑛 +s 1s ))))
69	52, 68	cbvrex2vw 3245	. . . . . . 7 ⊢ (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s )) ↔ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑛 +s 1s )))
70	46, 69	bitrdi 290	. . . . . 6 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s )) ↔ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑛 +s 1s ))))
71	42, 70	orbi12d 932	. . . . 5 ⊢ (𝑧 = 𝑤 → ((𝑧 = (𝑛 +s 1s ) ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s ))) ↔ (𝑤 = (𝑛 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑛 +s 1s )))))
72	41, 71	imbi12d 347	. . . 4 ⊢ (𝑧 = 𝑤 → ((((bday‘𝑧) ⊆ (bday‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑧) → (𝑧 = (𝑛 +s 1s ) ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s )))) ↔ (((bday‘𝑤) ⊆ (bday‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑤) → (𝑤 = (𝑛 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑛 +s 1s ))))))
73	72	cbvralvw 3240	. . 3 ⊢ (∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑧) → (𝑧 = (𝑛 +s 1s ) ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s (𝑛 +s 1s )))) ↔ ∀𝑤 ∈ No (((bday‘𝑤) ⊆ (bday‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑤) → (𝑤 = (𝑛 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑛 +s 1s )))))
74	37, 73	bitrdi 290	. 2 ⊢ (𝑚 = (𝑛 +s 1s ) → (∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚))) ↔ ∀𝑤 ∈ No (((bday‘𝑤) ⊆ (bday‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑤) → (𝑤 = (𝑛 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑛 +s 1s ))))))
75		fveq2 6885	. . . . . 6 ⊢ (𝑚 = 𝑁 → (bday‘𝑚) = (bday‘𝑁))
76	75	sseq2d 3940	. . . . 5 ⊢ (𝑚 = 𝑁 → ((bday‘𝑧) ⊆ (bday‘𝑚) ↔ (bday‘𝑧) ⊆ (bday‘𝑁)))
77	76	anbi1d 643	. . . 4 ⊢ (𝑚 = 𝑁 → (((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) ↔ ((bday‘𝑧) ⊆ (bday‘𝑁) ∧ 0s ≤s 𝑧)))
78		eqeq2 2772	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑧 = 𝑚 ↔ 𝑧 = 𝑁))
79		breq2 5106	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝑥 +s 𝑝) <s 𝑚 ↔ (𝑥 +s 𝑝) <s 𝑁))
80	79	3anbi3d 1470	. . . . . . 7 ⊢ (𝑚 = 𝑁 → ((𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚) ↔ (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
81	80	rexbidv 3186	. . . . . 6 ⊢ (𝑚 = 𝑁 → (∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚) ↔ ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
82	81	2rexbidv 3227	. . . . 5 ⊢ (𝑚 = 𝑁 → (∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚) ↔ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))
83	78, 82	orbi12d 932	. . . 4 ⊢ (𝑚 = 𝑁 → ((𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚)) ↔ (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))
84	77, 83	imbi12d 347	. . 3 ⊢ (𝑚 = 𝑁 → ((((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚))) ↔ (((bday‘𝑧) ⊆ (bday‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))))
85	84	ralbidv 3185	. 2 ⊢ (𝑚 = 𝑁 → (∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑚) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑚 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑚))) ↔ ∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁)))))
86		bday0b 28199	. . . . 5 ⊢ (𝑧 ∈ No → ((bday‘𝑧) = ∅ ↔ 𝑧 = 0s ))
87		orc 881	. . . . 5 ⊢ (𝑧 = 0s → (𝑧 = 0s ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 0s )))
88	86, 87	biimtrdi 256	. . . 4 ⊢ (𝑧 ∈ No → ((bday‘𝑧) = ∅ → (𝑧 = 0s ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 0s ))))
89	88	adantrd 497	. . 3 ⊢ (𝑧 ∈ No → (((bday‘𝑧) = ∅ ∧ 0s ≤s 𝑧) → (𝑧 = 0s ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 0s ))))
90	89	rgen 3078	. 2 ⊢ ∀𝑧 ∈ No (((bday‘𝑧) = ∅ ∧ 0s ≤s 𝑧) → (𝑧 = 0s ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 0s )))
91		simpl 488	. . . 4 ⊢ ((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑛) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑛 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑛)))) → 𝑛 ∈ ℕ0s)
92		simpr 490	. . . . 5 ⊢ ((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑛) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑛 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑛)))) → ∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑛) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑛 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑛))))
93	91, 92	bdayfinbndcbv 28852	. . . 4 ⊢ ((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑛) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑛 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑛)))) → ∀𝑡 ∈ No (((bday‘𝑡) ⊆ (bday‘𝑛) ∧ 0s ≤s 𝑡) → (𝑡 = 𝑛 ∨ ∃𝑐 ∈ ℕ0s ∃𝑑 ∈ ℕ0s ∃𝑟 ∈ ℕ0s (𝑡 = (𝑐 +s (𝑑 /su (2s↑s𝑟))) ∧ 𝑑 <s (2s↑s𝑟) ∧ (𝑐 +s 𝑟) <s 𝑛))))
94	91, 93	bdayfinbndlem1 28853	. . 3 ⊢ ((𝑛 ∈ ℕ0s ∧ ∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑛) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑛 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑛)))) → ∀𝑤 ∈ No (((bday‘𝑤) ⊆ (bday‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑤) → (𝑤 = (𝑛 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑛 +s 1s )))))
95	94	ex 418	. 2 ⊢ (𝑛 ∈ ℕ0s → (∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑛) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑛 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑛))) → ∀𝑤 ∈ No (((bday‘𝑤) ⊆ (bday‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑤) → (𝑤 = (𝑛 +s 1s ) ∨ ∃𝑎 ∈ ℕ0s ∃𝑏 ∈ ℕ0s ∃𝑞 ∈ ℕ0s (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞))) ∧ 𝑏 <s (2s↑s𝑞) ∧ (𝑎 +s 𝑞) <s (𝑛 +s 1s ))))))
96	15, 26, 74, 85, 90, 95	n0sind 28719	1 ⊢ (𝑁 ∈ ℕ0s → ∀𝑧 ∈ No (((bday‘𝑧) ⊆ (bday‘𝑁) ∧ 0s ≤s 𝑧) → (𝑧 = 𝑁 ∨ ∃𝑥 ∈ ℕ0s ∃𝑦 ∈ ℕ0s ∃𝑝 ∈ ℕ0s (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ∧ 𝑦 <s (2s↑s𝑝) ∧ (𝑥 +s 𝑝) <s 𝑁))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895  ∅c0 4278   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Nocsur 27997   <s clts 27998  bdaycbday 27999   ≤s cles 28101   0_s c0s 28191   1_s c1s 28192   +_s cadds 28345   /_su cdivs 28573  ℕ_0scn0s 28698  2_sc2s 28796  ↑_scexps 28798

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-dc 10524

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-nadd 8675  df-en 8974  df-dom 8975  df-fin 8977  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-1s 28194  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec 28324  df-norec2 28335  df-adds 28346  df-negs 28407  df-subs 28408  df-muls 28493  df-divs 28574  df-ons 28638  df-seqs 28670  df-n0s 28700  df-nns 28701  df-zs 28765  df-2s 28797  df-exps 28799

This theorem is used by:  bdayfinbnd  28855