Theorem bdayfinbndlem2 28627

Description: Lemma for bdayfinbnd 28628. 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 6882	. . . . . . . 8 ⊢ (𝑚 = 0s → ( bday ‘𝑚) = ( bday ‘ 0s ))
2		bday0 27970	. . . . . . . 8 ⊢ ( bday ‘ 0s ) = ∅
3	1, 2	eqtrdi 2820	. . . . . . 7 ⊢ (𝑚 = 0s → ( bday ‘𝑚) = ∅)
4	3	sseq2d 3975	. . . . . 6 ⊢ (𝑚 = 0s → (( bday ‘𝑧) ⊆ ( bday ‘𝑚) ↔ ( bday ‘𝑧) ⊆ ∅))
5		ss0b 4363	. . . . . 6 ⊢ (( bday ‘𝑧) ⊆ ∅ ↔ ( bday ‘𝑧) = ∅)
6	4, 5	bitrdi 290	. . . . 5 ⊢ (𝑚 = 0s → (( bday ‘𝑧) ⊆ ( bday ‘𝑚) ↔ ( bday ‘𝑧) = ∅))
7	6	anbi1d 642	. . . 4 ⊢ (𝑚 = 0s → ((( bday ‘𝑧) ⊆ ( bday ‘𝑚) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑧) = ∅ ∧ 0s ≤s 𝑧)))
8		eqeq2 2781	. . . . 5 ⊢ (𝑚 = 0s → (𝑧 = 𝑚 ↔ 𝑧 = 0s ))
9		breq2 5115	. . . . . . . 8 ⊢ (𝑚 = 0s → ((𝑥 +s 𝑝) <s 𝑚 ↔ (𝑥 +s 𝑝) <s 0s ))
10	9	3anbi3d 1468	. . . . . . 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 3195	. . . . . 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 3236	. . . . 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 931	. . . 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 3194	. 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 6882	. . . . . 6 ⊢ (𝑚 = 𝑛 → ( bday ‘𝑚) = ( bday ‘𝑛))
17	16	sseq2d 3975	. . . . 5 ⊢ (𝑚 = 𝑛 → (( bday ‘𝑧) ⊆ ( bday ‘𝑚) ↔ ( bday ‘𝑧) ⊆ ( bday ‘𝑛)))
18	17	anbi1d 642	. . . 4 ⊢ (𝑚 = 𝑛 → ((( bday ‘𝑧) ⊆ ( bday ‘𝑚) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑧) ⊆ ( bday ‘𝑛) ∧ 0s ≤s 𝑧)))
19		eqeq2 2781	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑧 = 𝑚 ↔ 𝑧 = 𝑛))
20		breq2 5115	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝑥 +s 𝑝) <s 𝑚 ↔ (𝑥 +s 𝑝) <s 𝑛))
21	20	3anbi3d 1468	. . . . . . 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 3195	. . . . . 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 3236	. . . . 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 931	. . . 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 3194	. 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 6882	. . . . . . 7 ⊢ (𝑚 = (𝑛 +s 1s ) → ( bday ‘𝑚) = ( bday ‘(𝑛 +s 1s )))
28	27	sseq2d 3975	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (( bday ‘𝑧) ⊆ ( bday ‘𝑚) ↔ ( bday ‘𝑧) ⊆ ( bday ‘(𝑛 +s 1s ))))
29	28	anbi1d 642	. . . . 5 ⊢ (𝑚 = (𝑛 +s 1s ) → ((( bday ‘𝑧) ⊆ ( bday ‘𝑚) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑧) ⊆ ( bday ‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑧)))
30		eqeq2 2781	. . . . . 6 ⊢ (𝑚 = (𝑛 +s 1s ) → (𝑧 = 𝑚 ↔ 𝑧 = (𝑛 +s 1s )))
31		breq2 5115	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 +s 1s ) → ((𝑥 +s 𝑝) <s 𝑚 ↔ (𝑥 +s 𝑝) <s (𝑛 +s 1s )))
32	31	3anbi3d 1468	. . . . . . . 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 3195	. . . . . . 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 3236	. . . . . 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 931	. . . . 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 3194	. . 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 6882	. . . . . . 7 ⊢ (𝑧 = 𝑤 → ( bday ‘𝑧) = ( bday ‘𝑤))
39	38	sseq1d 3974	. . . . . 6 ⊢ (𝑧 = 𝑤 → (( bday ‘𝑧) ⊆ ( bday ‘(𝑛 +s 1s )) ↔ ( bday ‘𝑤) ⊆ ( bday ‘(𝑛 +s 1s ))))
40		breq2 5115	. . . . . 6 ⊢ (𝑧 = 𝑤 → ( 0s ≤s 𝑧 ↔ 0s ≤s 𝑤))
41	39, 40	anbi12d 643	. . . . 5 ⊢ (𝑧 = 𝑤 → ((( bday ‘𝑧) ⊆ ( bday ‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑤) ⊆ ( bday ‘(𝑛 +s 1s )) ∧ 0s ≤s 𝑤)))
42		eqeq1 2773	. . . . . 6 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑛 +s 1s ) ↔ 𝑤 = (𝑛 +s 1s )))
43		eqeq1 2773	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → (𝑧 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝)))))
44	43	3anbi1d 1466	. . . . . . . . 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 3195	. . . . . . . 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 3236	. . . . . . 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 7418	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑥 +s (𝑦 /su (2s↑s𝑝))) = (𝑎 +s (𝑦 /su (2s↑s𝑝))))
48	47	eqeq2d 2780	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝑤 = (𝑥 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝)))))
49		oveq1 7418	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑥 +s 𝑝) = (𝑎 +s 𝑝))
50	49	breq1d 5121	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ((𝑥 +s 𝑝) <s (𝑛 +s 1s ) ↔ (𝑎 +s 𝑝) <s (𝑛 +s 1s )))
51	48, 50	3anbi13d 1464	. . . . . . . . 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 3195	. . . . . . . 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 7418	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑏 → (𝑦 /su (2s↑s𝑝)) = (𝑏 /su (2s↑s𝑝)))
54	53	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → (𝑎 +s (𝑦 /su (2s↑s𝑝))) = (𝑎 +s (𝑏 /su (2s↑s𝑝))))
55	54	eqeq2d 2780	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (𝑤 = (𝑎 +s (𝑦 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝)))))
56		breq1 5114	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (𝑦 <s (2s↑s𝑝) ↔ 𝑏 <s (2s↑s𝑝)))
57	55, 56	3anbi12d 1463	. . . . . . . . . 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 3195	. . . . . . . . 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 7419	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑞 → (2s↑s𝑝) = (2s↑s𝑞))
60	59	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑞 → (𝑏 /su (2s↑s𝑝)) = (𝑏 /su (2s↑s𝑞)))
61	60	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑞 → (𝑎 +s (𝑏 /su (2s↑s𝑝))) = (𝑎 +s (𝑏 /su (2s↑s𝑞))))
62	61	eqeq2d 2780	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑞 → (𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑝))) ↔ 𝑤 = (𝑎 +s (𝑏 /su (2s↑s𝑞)))))
63	59	breq2d 5123	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑞 → (𝑏 <s (2s↑s𝑝) ↔ 𝑏 <s (2s↑s𝑞)))
64		oveq2 7419	. . . . . . . . . . . 12 ⊢ (𝑝 = 𝑞 → (𝑎 +s 𝑝) = (𝑎 +s 𝑞))
65	64	breq1d 5121	. . . . . . . . . . 11 ⊢ (𝑝 = 𝑞 → ((𝑎 +s 𝑝) <s (𝑛 +s 1s ) ↔ (𝑎 +s 𝑞) <s (𝑛 +s 1s )))
66	62, 63, 65	3anbi123d 1462	. . . . . . . . . 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 3250	. . . . . . . . 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 3254	. . . . . . 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 931	. . . . 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 3249	. . 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 6882	. . . . . 6 ⊢ (𝑚 = 𝑁 → ( bday ‘𝑚) = ( bday ‘𝑁))
76	75	sseq2d 3975	. . . . 5 ⊢ (𝑚 = 𝑁 → (( bday ‘𝑧) ⊆ ( bday ‘𝑚) ↔ ( bday ‘𝑧) ⊆ ( bday ‘𝑁)))
77	76	anbi1d 642	. . . 4 ⊢ (𝑚 = 𝑁 → ((( bday ‘𝑧) ⊆ ( bday ‘𝑚) ∧ 0s ≤s 𝑧) ↔ (( bday ‘𝑧) ⊆ ( bday ‘𝑁) ∧ 0s ≤s 𝑧)))
78		eqeq2 2781	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑧 = 𝑚 ↔ 𝑧 = 𝑁))
79		breq2 5115	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → ((𝑥 +s 𝑝) <s 𝑚 ↔ (𝑥 +s 𝑝) <s 𝑁))
80	79	3anbi3d 1468	. . . . . . 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 3195	. . . . . 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 3236	. . . . 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 931	. . . 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 3194	. 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 27972	. . . . 5 ⊢ (𝑧 ∈ No → (( bday ‘𝑧) = ∅ ↔ 𝑧 = 0s ))
87		orc 880	. . . . 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 496	. . 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 3087	. 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 487	. . . 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 489	. . . . 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 28625	. . . 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 28626	. . 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 417	. 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 28492	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))))

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095   ⊆ wss 3911  ∅c0 4292   class class class wbr 5111  ‘cfv 6537  (class class class)co 7411   No csur 27770   <s clts 27771   bday cbday 27772   ≤s cles 27874   0_s c0s 27964   1_s c1s 27965   +_s cadds 28118   /_su cdivs 28346  ℕ_0scn0s 28471  2_sc2s 28569  ↑_scexps 28571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-dc 10430  ax-ac2 10447

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-oadd 8457  df-nadd 8652  df-er 8694  df-map 8826  df-en 8944  df-dom 8945  df-fin 8947  df-card 9925  df-acn 9928  df-ac 10100  df-no 27773  df-lts 27774  df-bday 27775  df-les 27875  df-slts 27917  df-cuts 27919  df-0s 27966  df-1s 27967  df-made 27986  df-old 27987  df-left 27989  df-right 27990  df-norec 28097  df-norec2 28108  df-adds 28119  df-negs 28180  df-subs 28181  df-muls 28266  df-divs 28347  df-ons 28411  df-seqs 28443  df-n0s 28473  df-nns 28474  df-zs 28538  df-2s 28570  df-exps 28572

This theorem is referenced by:  bdayfinbnd  28628