Theorem addonbday 28450

Description: The birthday of the sum of two ordinals is the natural sum of their birthdays. (Contributed by Scott Fenton, 22-Feb-2026.)

Assertion

Ref	Expression
addonbday	⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( bday ‘(𝐴 +s 𝐵)) = (( bday ‘𝐴) +no ( bday ‘𝐵)))

Proof of Theorem addonbday

Dummy variables 𝑥 𝑥_𝑂 𝑦 𝑦_𝑂 𝑎 𝑝 𝑞 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onno 28426	. . 3 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
2		onno 28426	. . 3 ⊢ (𝐵 ∈ Ons → 𝐵 ∈ No )
3		addbday 28189	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → ( bday ‘(𝐴 +s 𝐵)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)))
4	1, 2, 3	syl2an 607	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( bday ‘(𝐴 +s 𝐵)) ⊆ (( bday ‘𝐴) +no ( bday ‘𝐵)))
5		fveq2 6883	. . . . 5 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘𝑥) = ( bday ‘𝑥𝑂))
6	5	oveq1d 7427	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)))
7		fvoveq1 7435	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘(𝑥 +s 𝑦)) = ( bday ‘(𝑥𝑂 +s 𝑦)))
8	6, 7	sseq12d 3971	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((( bday ‘𝑥) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥 +s 𝑦)) ↔ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))))
9		fveq2 6883	. . . . 5 ⊢ (𝑦 = 𝑦𝑂 → ( bday ‘𝑦) = ( bday ‘𝑦𝑂))
10	9	oveq2d 7428	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)))
11		oveq2 7420	. . . . 5 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
12	11	fveq2d 6887	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → ( bday ‘(𝑥𝑂 +s 𝑦)) = ( bday ‘(𝑥𝑂 +s 𝑦𝑂)))
13	10, 12	sseq12d 3971	. . 3 ⊢ (𝑦 = 𝑦𝑂 → ((( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦)) ↔ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))))
14	5	oveq1d 7427	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) = (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)))
15		fvoveq1 7435	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → ( bday ‘(𝑥 +s 𝑦𝑂)) = ( bday ‘(𝑥𝑂 +s 𝑦𝑂)))
16	14, 15	sseq12d 3971	. . 3 ⊢ (𝑥 = 𝑥𝑂 → ((( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂)) ↔ (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))))
17		fveq2 6883	. . . . 5 ⊢ (𝑥 = 𝐴 → ( bday ‘𝑥) = ( bday ‘𝐴))
18	17	oveq1d 7427	. . . 4 ⊢ (𝑥 = 𝐴 → (( bday ‘𝑥) +no ( bday ‘𝑦)) = (( bday ‘𝐴) +no ( bday ‘𝑦)))
19		fvoveq1 7435	. . . 4 ⊢ (𝑥 = 𝐴 → ( bday ‘(𝑥 +s 𝑦)) = ( bday ‘(𝐴 +s 𝑦)))
20	18, 19	sseq12d 3971	. . 3 ⊢ (𝑥 = 𝐴 → ((( bday ‘𝑥) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥 +s 𝑦)) ↔ (( bday ‘𝐴) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝐴 +s 𝑦))))
21		fveq2 6883	. . . . 5 ⊢ (𝑦 = 𝐵 → ( bday ‘𝑦) = ( bday ‘𝐵))
22	21	oveq2d 7428	. . . 4 ⊢ (𝑦 = 𝐵 → (( bday ‘𝐴) +no ( bday ‘𝑦)) = (( bday ‘𝐴) +no ( bday ‘𝐵)))
23		oveq2 7420	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
24	23	fveq2d 6887	. . . 4 ⊢ (𝑦 = 𝐵 → ( bday ‘(𝐴 +s 𝑦)) = ( bday ‘(𝐴 +s 𝐵)))
25	22, 24	sseq12d 3971	. . 3 ⊢ (𝑦 = 𝐵 → ((( bday ‘𝐴) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝐴 +s 𝑦)) ↔ (( bday ‘𝐴) +no ( bday ‘𝐵)) ⊆ ( bday ‘(𝐴 +s 𝐵))))
26		bdayon 27923	. . . . . 6 ⊢ ( bday ‘𝑥) ∈ On
27		bdayon 27923	. . . . . 6 ⊢ ( bday ‘𝑦) ∈ On
28		naddov2 8666	. . . . . 6 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘𝑦) ∈ On) → (( bday ‘𝑥) +no ( bday ‘𝑦)) = ∩ {𝑎 ∈ On ∣ (∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ 𝑎 ∧ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ 𝑎)})
29	26, 27, 28	mp2an 704	. . . . 5 ⊢ (( bday ‘𝑥) +no ( bday ‘𝑦)) = ∩ {𝑎 ∈ On ∣ (∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ 𝑎 ∧ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ 𝑎)}
30	27	oneli 6478	. . . . . . . . . 10 ⊢ (𝑞 ∈ ( bday ‘𝑦) → 𝑞 ∈ On)
31		breq1 5113	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦𝑂 = (◡( bday ↾ Ons)‘𝑞) → (𝑦𝑂 <s 𝑦 ↔ (◡( bday ↾ Ons)‘𝑞) <s 𝑦))
32		fveq2 6883	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦𝑂 = (◡( bday ↾ Ons)‘𝑞) → ( bday ‘𝑦𝑂) = ( bday ‘(◡( bday ↾ Ons)‘𝑞)))
33	32	oveq2d 7428	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦𝑂 = (◡( bday ↾ Ons)‘𝑞) → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) = (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))))
34		oveq2 7420	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦𝑂 = (◡( bday ↾ Ons)‘𝑞) → (𝑥 +s 𝑦𝑂) = (𝑥 +s (◡( bday ↾ Ons)‘𝑞)))
35	34	fveq2d 6887	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦𝑂 = (◡( bday ↾ Ons)‘𝑞) → ( bday ‘(𝑥 +s 𝑦𝑂)) = ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞))))
36	33, 35	sseq12d 3971	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦𝑂 = (◡( bday ↾ Ons)‘𝑞) → ((( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂)) ↔ (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ⊆ ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞)))))
37	31, 36	imbi12d 347	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦𝑂 = (◡( bday ↾ Ons)‘𝑞) → ((𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))) ↔ ((◡( bday ↾ Ons)‘𝑞) <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ⊆ ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞))))))
38		simplr3 1236	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))
39		oniso 28442	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( bday ↾ Ons) Isom <s , E (Ons, On)
40		isof1o 7323	. . . . . . . . . . . . . . . . . . . 20 ⊢ (( bday ↾ Ons) Isom <s , E (Ons, On) → ( bday ↾ Ons):Ons–1-1-onto→On)
41	39, 40	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ( bday ↾ Ons):Ons–1-1-onto→On
42		f1ocnvdm 7285	. . . . . . . . . . . . . . . . . . 19 ⊢ ((( bday ↾ Ons):Ons–1-1-onto→On ∧ 𝑞 ∈ On) → (◡( bday ↾ Ons)‘𝑞) ∈ Ons)
43	41, 42	mpan 702	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 ∈ On → (◡( bday ↾ Ons)‘𝑞) ∈ Ons)
44	43	adantl 486	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → (◡( bday ↾ Ons)‘𝑞) ∈ Ons)
45	37, 38, 44	rspcdva 3583	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → ((◡( bday ↾ Ons)‘𝑞) <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ⊆ ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞)))))
46	45	impr 459	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ (𝑞 ∈ On ∧ (◡( bday ↾ Ons)‘𝑞) <s 𝑦)) → (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ⊆ ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞))))
47		onno 28426	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((◡( bday ↾ Ons)‘𝑞) ∈ Ons → (◡( bday ↾ Ons)‘𝑞) ∈ No )
48	44, 47	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → (◡( bday ↾ Ons)‘𝑞) ∈ No )
49		simpllr 787	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → 𝑦 ∈ Ons)
50		onno 28426	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ Ons → 𝑦 ∈ No )
51	49, 50	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → 𝑦 ∈ No )
52		simplll 786	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → 𝑥 ∈ Ons)
53		onno 28426	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ Ons → 𝑥 ∈ No )
54	52, 53	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → 𝑥 ∈ No )
55	48, 51, 54	ltadds2d 28168	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → ((◡( bday ↾ Ons)‘𝑞) <s 𝑦 ↔ (𝑥 +s (◡( bday ↾ Ons)‘𝑞)) <s (𝑥 +s 𝑦)))
56		onaddscl 28448	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ Ons ∧ (◡( bday ↾ Ons)‘𝑞) ∈ Ons) → (𝑥 +s (◡( bday ↾ Ons)‘𝑞)) ∈ Ons)
57	52, 44, 56	syl2anc 595	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → (𝑥 +s (◡( bday ↾ Ons)‘𝑞)) ∈ Ons)
58		onaddscl 28448	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (𝑥 +s 𝑦) ∈ Ons)
59	58	ad2antrr 738	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → (𝑥 +s 𝑦) ∈ Ons)
60		onlts 28438	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑥 +s (◡( bday ↾ Ons)‘𝑞)) ∈ Ons ∧ (𝑥 +s 𝑦) ∈ Ons) → ((𝑥 +s (◡( bday ↾ Ons)‘𝑞)) <s (𝑥 +s 𝑦) ↔ ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦))))
61	57, 59, 60	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → ((𝑥 +s (◡( bday ↾ Ons)‘𝑞)) <s (𝑥 +s 𝑦) ↔ ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦))))
62	55, 61	bitrd 282	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → ((◡( bday ↾ Ons)‘𝑞) <s 𝑦 ↔ ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦))))
63	62	biimpd 232	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → ((◡( bday ↾ Ons)‘𝑞) <s 𝑦 → ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦))))
64	63	impr 459	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ (𝑞 ∈ On ∧ (◡( bday ↾ Ons)‘𝑞) <s 𝑦)) → ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦)))
65		bdayon 27923	. . . . . . . . . . . . . . . . . 18 ⊢ ( bday ‘(◡( bday ↾ Ons)‘𝑞)) ∈ On
66		naddcl 8664	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ‘𝑥) ∈ On ∧ ( bday ‘(◡( bday ↾ Ons)‘𝑞)) ∈ On) → (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ∈ On)
67	26, 65, 66	mp2an 704	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ∈ On
68	67	onordi 6476	. . . . . . . . . . . . . . . 16 ⊢ Ord (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞)))
69		bdayon 27923	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘(𝑥 +s 𝑦)) ∈ On
70	69	onordi 6476	. . . . . . . . . . . . . . . 16 ⊢ Ord ( bday ‘(𝑥 +s 𝑦))
71		ordtr2 6408	. . . . . . . . . . . . . . . 16 ⊢ ((Ord (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ∧ Ord ( bday ‘(𝑥 +s 𝑦))) → (((( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ⊆ ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞))) ∧ ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦))) → (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦))))
72	68, 70, 71	mp2an 704	. . . . . . . . . . . . . . 15 ⊢ (((( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ⊆ ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞))) ∧ ( bday ‘(𝑥 +s (◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦))) → (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦)))
73	46, 64, 72	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ (𝑞 ∈ On ∧ (◡( bday ↾ Ons)‘𝑞) <s 𝑦)) → (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦)))
74	73	expr 461	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → ((◡( bday ↾ Ons)‘𝑞) <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦))))
75	43	fvresd 6903	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∈ On → (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑞)) = ( bday ‘(◡( bday ↾ Ons)‘𝑞)))
76	75	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑞)) = ( bday ‘(◡( bday ↾ Ons)‘𝑞)))
77	76	oveq2d 7428	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → (( bday ‘𝑥) +no (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑞))) = (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))))
78	77	eleq1d 2848	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → ((( bday ‘𝑥) +no (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦)) ↔ (( bday ‘𝑥) +no ( bday ‘(◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦))))
79	74, 78	sylibrd 262	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → ((◡( bday ↾ Ons)‘𝑞) <s 𝑦 → (( bday ‘𝑥) +no (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦))))
80		onlts 28438	. . . . . . . . . . . . . 14 ⊢ (((◡( bday ↾ Ons)‘𝑞) ∈ Ons ∧ 𝑦 ∈ Ons) → ((◡( bday ↾ Ons)‘𝑞) <s 𝑦 ↔ ( bday ‘(◡( bday ↾ Ons)‘𝑞)) ∈ ( bday ‘𝑦)))
81	44, 49, 80	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → ((◡( bday ↾ Ons)‘𝑞) <s 𝑦 ↔ ( bday ‘(◡( bday ↾ Ons)‘𝑞)) ∈ ( bday ‘𝑦)))
82		f1ocnvfv2 7277	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ↾ Ons):Ons–1-1-onto→On ∧ 𝑞 ∈ On) → (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑞)) = 𝑞)
83	41, 82	mpan 702	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∈ On → (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑞)) = 𝑞)
84	75, 83	eqtr3d 2800	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ On → ( bday ‘(◡( bday ↾ Ons)‘𝑞)) = 𝑞)
85	84	eleq1d 2848	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ On → (( bday ‘(◡( bday ↾ Ons)‘𝑞)) ∈ ( bday ‘𝑦) ↔ 𝑞 ∈ ( bday ‘𝑦)))
86	85	adantl 486	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → (( bday ‘(◡( bday ↾ Ons)‘𝑞)) ∈ ( bday ‘𝑦) ↔ 𝑞 ∈ ( bday ‘𝑦)))
87	81, 86	bitrd 282	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → ((◡( bday ↾ Ons)‘𝑞) <s 𝑦 ↔ 𝑞 ∈ ( bday ‘𝑦)))
88	83	oveq2d 7428	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ On → (( bday ‘𝑥) +no (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑞))) = (( bday ‘𝑥) +no 𝑞))
89	88	eleq1d 2848	. . . . . . . . . . . . 13 ⊢ (𝑞 ∈ On → ((( bday ‘𝑥) +no (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦)) ↔ (( bday ‘𝑥) +no 𝑞) ∈ ( bday ‘(𝑥 +s 𝑦))))
90	89	adantl 486	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → ((( bday ‘𝑥) +no (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑞))) ∈ ( bday ‘(𝑥 +s 𝑦)) ↔ (( bday ‘𝑥) +no 𝑞) ∈ ( bday ‘(𝑥 +s 𝑦))))
91	79, 87, 90	3imtr3d 296	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑞 ∈ On) → (𝑞 ∈ ( bday ‘𝑦) → (( bday ‘𝑥) +no 𝑞) ∈ ( bday ‘(𝑥 +s 𝑦))))
92	91	ex 417	. . . . . . . . . 10 ⊢ (((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) → (𝑞 ∈ On → (𝑞 ∈ ( bday ‘𝑦) → (( bday ‘𝑥) +no 𝑞) ∈ ( bday ‘(𝑥 +s 𝑦)))))
93	30, 92	syl5 35	. . . . . . . . 9 ⊢ (((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) → (𝑞 ∈ ( bday ‘𝑦) → (𝑞 ∈ ( bday ‘𝑦) → (( bday ‘𝑥) +no 𝑞) ∈ ( bday ‘(𝑥 +s 𝑦)))))
94	93	pm2.43d 54	. . . . . . . 8 ⊢ (((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) → (𝑞 ∈ ( bday ‘𝑦) → (( bday ‘𝑥) +no 𝑞) ∈ ( bday ‘(𝑥 +s 𝑦))))
95	94	ralrimiv 3156	. . . . . . 7 ⊢ (((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) → ∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ ( bday ‘(𝑥 +s 𝑦)))
96	26	oneli 6478	. . . . . . . . . 10 ⊢ (𝑝 ∈ ( bday ‘𝑥) → 𝑝 ∈ On)
97		breq1 5113	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = (◡( bday ↾ Ons)‘𝑝) → (𝑥𝑂 <s 𝑥 ↔ (◡( bday ↾ Ons)‘𝑝) <s 𝑥))
98		fveq2 6883	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = (◡( bday ↾ Ons)‘𝑝) → ( bday ‘𝑥𝑂) = ( bday ‘(◡( bday ↾ Ons)‘𝑝)))
99	98	oveq1d 7427	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = (◡( bday ↾ Ons)‘𝑝) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) = (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)))
100		fvoveq1 7435	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = (◡( bday ↾ Ons)‘𝑝) → ( bday ‘(𝑥𝑂 +s 𝑦)) = ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦)))
101	99, 100	sseq12d 3971	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = (◡( bday ↾ Ons)‘𝑝) → ((( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦)) ↔ (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ⊆ ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦))))
102	97, 101	imbi12d 347	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑂 = (◡( bday ↾ Ons)‘𝑝) → ((𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ↔ ((◡( bday ↾ Ons)‘𝑝) <s 𝑥 → (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ⊆ ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦)))))
103		simplr2 1235	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))))
104		f1ocnvdm 7285	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ↾ Ons):Ons–1-1-onto→On ∧ 𝑝 ∈ On) → (◡( bday ↾ Ons)‘𝑝) ∈ Ons)
105	41, 104	mpan 702	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ On → (◡( bday ↾ Ons)‘𝑝) ∈ Ons)
106	105	adantl 486	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → (◡( bday ↾ Ons)‘𝑝) ∈ Ons)
107	102, 103, 106	rspcdva 3583	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → ((◡( bday ↾ Ons)‘𝑝) <s 𝑥 → (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ⊆ ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦))))
108	107	impr 459	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ (𝑝 ∈ On ∧ (◡( bday ↾ Ons)‘𝑝) <s 𝑥)) → (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ⊆ ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦)))
109		onno 28426	. . . . . . . . . . . . . . . . . . 19 ⊢ ((◡( bday ↾ Ons)‘𝑝) ∈ Ons → (◡( bday ↾ Ons)‘𝑝) ∈ No )
110	106, 109	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → (◡( bday ↾ Ons)‘𝑝) ∈ No )
111		simplll 786	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → 𝑥 ∈ Ons)
112	111, 53	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → 𝑥 ∈ No )
113		simpllr 787	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → 𝑦 ∈ Ons)
114	113, 50	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → 𝑦 ∈ No )
115	110, 112, 114	ltadds1d 28169	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → ((◡( bday ↾ Ons)‘𝑝) <s 𝑥 ↔ ((◡( bday ↾ Ons)‘𝑝) +s 𝑦) <s (𝑥 +s 𝑦)))
116		onaddscl 28448	. . . . . . . . . . . . . . . . . . 19 ⊢ (((◡( bday ↾ Ons)‘𝑝) ∈ Ons ∧ 𝑦 ∈ Ons) → ((◡( bday ↾ Ons)‘𝑝) +s 𝑦) ∈ Ons)
117	106, 113, 116	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → ((◡( bday ↾ Ons)‘𝑝) +s 𝑦) ∈ Ons)
118	58	ad2antrr 738	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → (𝑥 +s 𝑦) ∈ Ons)
119		onlts 28438	. . . . . . . . . . . . . . . . . 18 ⊢ ((((◡( bday ↾ Ons)‘𝑝) +s 𝑦) ∈ Ons ∧ (𝑥 +s 𝑦) ∈ Ons) → (((◡( bday ↾ Ons)‘𝑝) +s 𝑦) <s (𝑥 +s 𝑦) ↔ ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
120	117, 118, 119	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → (((◡( bday ↾ Ons)‘𝑝) +s 𝑦) <s (𝑥 +s 𝑦) ↔ ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
121	115, 120	bitrd 282	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → ((◡( bday ↾ Ons)‘𝑝) <s 𝑥 ↔ ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
122	121	biimpd 232	. . . . . . . . . . . . . . 15 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → ((◡( bday ↾ Ons)‘𝑝) <s 𝑥 → ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
123	122	impr 459	. . . . . . . . . . . . . 14 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ (𝑝 ∈ On ∧ (◡( bday ↾ Ons)‘𝑝) <s 𝑥)) → ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦)))
124		bdayon 27923	. . . . . . . . . . . . . . . . 17 ⊢ ( bday ‘(◡( bday ↾ Ons)‘𝑝)) ∈ On
125		naddcl 8664	. . . . . . . . . . . . . . . . 17 ⊢ ((( bday ‘(◡( bday ↾ Ons)‘𝑝)) ∈ On ∧ ( bday ‘𝑦) ∈ On) → (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ∈ On)
126	124, 27, 125	mp2an 704	. . . . . . . . . . . . . . . 16 ⊢ (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ∈ On
127	126	onordi 6476	. . . . . . . . . . . . . . 15 ⊢ Ord (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦))
128		ordtr2 6408	. . . . . . . . . . . . . . 15 ⊢ ((Ord (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ∧ Ord ( bday ‘(𝑥 +s 𝑦))) → (((( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ⊆ ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦)) ∧ ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))) → (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
129	127, 70, 128	mp2an 704	. . . . . . . . . . . . . 14 ⊢ (((( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ⊆ ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦)) ∧ ( bday ‘((◡( bday ↾ Ons)‘𝑝) +s 𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))) → (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦)))
130	108, 123, 129	syl2anc 595	. . . . . . . . . . . . 13 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ (𝑝 ∈ On ∧ (◡( bday ↾ Ons)‘𝑝) <s 𝑥)) → (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦)))
131	130	expr 461	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → ((◡( bday ↾ Ons)‘𝑝) <s 𝑥 → (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
132		onlts 28438	. . . . . . . . . . . . . . . 16 ⊢ (((◡( bday ↾ Ons)‘𝑝) ∈ Ons ∧ 𝑥 ∈ Ons) → ((◡( bday ↾ Ons)‘𝑝) <s 𝑥 ↔ ( bday ‘(◡( bday ↾ Ons)‘𝑝)) ∈ ( bday ‘𝑥)))
133	105, 132	sylan 591	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ On ∧ 𝑥 ∈ Ons) → ((◡( bday ↾ Ons)‘𝑝) <s 𝑥 ↔ ( bday ‘(◡( bday ↾ Ons)‘𝑝)) ∈ ( bday ‘𝑥)))
134	133	ancoms 463	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Ons ∧ 𝑝 ∈ On) → ((◡( bday ↾ Ons)‘𝑝) <s 𝑥 ↔ ( bday ‘(◡( bday ↾ Ons)‘𝑝)) ∈ ( bday ‘𝑥)))
135	105	fvresd 6903	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ On → (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑝)) = ( bday ‘(◡( bday ↾ Ons)‘𝑝)))
136		f1ocnvfv2 7277	. . . . . . . . . . . . . . . . . 18 ⊢ ((( bday ↾ Ons):Ons–1-1-onto→On ∧ 𝑝 ∈ On) → (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑝)) = 𝑝)
137	41, 136	mpan 702	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ On → (( bday ↾ Ons)‘(◡( bday ↾ Ons)‘𝑝)) = 𝑝)
138	135, 137	eqtr3d 2800	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ On → ( bday ‘(◡( bday ↾ Ons)‘𝑝)) = 𝑝)
139	138	eleq1d 2848	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ On → (( bday ‘(◡( bday ↾ Ons)‘𝑝)) ∈ ( bday ‘𝑥) ↔ 𝑝 ∈ ( bday ‘𝑥)))
140	139	adantl 486	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Ons ∧ 𝑝 ∈ On) → (( bday ‘(◡( bday ↾ Ons)‘𝑝)) ∈ ( bday ‘𝑥) ↔ 𝑝 ∈ ( bday ‘𝑥)))
141	134, 140	bitrd 282	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ Ons ∧ 𝑝 ∈ On) → ((◡( bday ↾ Ons)‘𝑝) <s 𝑥 ↔ 𝑝 ∈ ( bday ‘𝑥)))
142	141	ad4ant14 764	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → ((◡( bday ↾ Ons)‘𝑝) <s 𝑥 ↔ 𝑝 ∈ ( bday ‘𝑥)))
143	138	oveq1d 7427	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ On → (( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) = (𝑝 +no ( bday ‘𝑦)))
144	143	eleq1d 2848	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ On → ((( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦)) ↔ (𝑝 +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
145	144	adantl 486	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → ((( bday ‘(◡( bday ↾ Ons)‘𝑝)) +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦)) ↔ (𝑝 +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
146	131, 142, 145	3imtr3d 296	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) ∧ 𝑝 ∈ On) → (𝑝 ∈ ( bday ‘𝑥) → (𝑝 +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
147	146	ex 417	. . . . . . . . . 10 ⊢ (((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) → (𝑝 ∈ On → (𝑝 ∈ ( bday ‘𝑥) → (𝑝 +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦)))))
148	96, 147	syl5 35	. . . . . . . . 9 ⊢ (((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) → (𝑝 ∈ ( bday ‘𝑥) → (𝑝 ∈ ( bday ‘𝑥) → (𝑝 +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦)))))
149	148	pm2.43d 54	. . . . . . . 8 ⊢ (((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) → (𝑝 ∈ ( bday ‘𝑥) → (𝑝 +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
150	149	ralrimiv 3156	. . . . . . 7 ⊢ (((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) → ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦)))
151		eleq2 2852	. . . . . . . . . . 11 ⊢ (𝑎 = ( bday ‘(𝑥 +s 𝑦)) → ((( bday ‘𝑥) +no 𝑞) ∈ 𝑎 ↔ (( bday ‘𝑥) +no 𝑞) ∈ ( bday ‘(𝑥 +s 𝑦))))
152	151	ralbidv 3188	. . . . . . . . . 10 ⊢ (𝑎 = ( bday ‘(𝑥 +s 𝑦)) → (∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ 𝑎 ↔ ∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ ( bday ‘(𝑥 +s 𝑦))))
153		eleq2 2852	. . . . . . . . . . 11 ⊢ (𝑎 = ( bday ‘(𝑥 +s 𝑦)) → ((𝑝 +no ( bday ‘𝑦)) ∈ 𝑎 ↔ (𝑝 +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
154	153	ralbidv 3188	. . . . . . . . . 10 ⊢ (𝑎 = ( bday ‘(𝑥 +s 𝑦)) → (∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ 𝑎 ↔ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
155	152, 154	anbi12d 643	. . . . . . . . 9 ⊢ (𝑎 = ( bday ‘(𝑥 +s 𝑦)) → ((∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ 𝑎 ∧ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ 𝑎) ↔ (∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ ( bday ‘(𝑥 +s 𝑦)) ∧ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦)))))
156	155	elrab3 3652	. . . . . . . 8 ⊢ (( bday ‘(𝑥 +s 𝑦)) ∈ On → (( bday ‘(𝑥 +s 𝑦)) ∈ {𝑎 ∈ On ∣ (∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ 𝑎 ∧ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ 𝑎)} ↔ (∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ ( bday ‘(𝑥 +s 𝑦)) ∧ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦)))))
157	69, 156	ax-mp 5	. . . . . . 7 ⊢ (( bday ‘(𝑥 +s 𝑦)) ∈ {𝑎 ∈ On ∣ (∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ 𝑎 ∧ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ 𝑎)} ↔ (∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ ( bday ‘(𝑥 +s 𝑦)) ∧ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ ( bday ‘(𝑥 +s 𝑦))))
158	95, 150, 157	sylanbrc 594	. . . . . 6 ⊢ (((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) → ( bday ‘(𝑥 +s 𝑦)) ∈ {𝑎 ∈ On ∣ (∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ 𝑎 ∧ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ 𝑎)})
159		intss1 4929	. . . . . 6 ⊢ (( bday ‘(𝑥 +s 𝑦)) ∈ {𝑎 ∈ On ∣ (∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ 𝑎 ∧ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ 𝑎)} → ∩ {𝑎 ∈ On ∣ (∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ 𝑎 ∧ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ 𝑎)} ⊆ ( bday ‘(𝑥 +s 𝑦)))
160	158, 159	syl 18	. . . . 5 ⊢ (((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) → ∩ {𝑎 ∈ On ∣ (∀𝑞 ∈ ( bday ‘𝑦)(( bday ‘𝑥) +no 𝑞) ∈ 𝑎 ∧ ∀𝑝 ∈ ( bday ‘𝑥)(𝑝 +no ( bday ‘𝑦)) ∈ 𝑎)} ⊆ ( bday ‘(𝑥 +s 𝑦)))
161	29, 160	eqsstrid 3976	. . . 4 ⊢ (((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) ∧ (∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂))))) → (( bday ‘𝑥) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥 +s 𝑦)))
162	161	ex 417	. . 3 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → ((∀𝑥𝑂 ∈ Ons ∀𝑦𝑂 ∈ Ons ((𝑥𝑂 <s 𝑥 ∧ 𝑦𝑂 <s 𝑦) → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦𝑂))) ∧ ∀𝑥𝑂 ∈ Ons (𝑥𝑂 <s 𝑥 → (( bday ‘𝑥𝑂) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥𝑂 +s 𝑦))) ∧ ∀𝑦𝑂 ∈ Ons (𝑦𝑂 <s 𝑦 → (( bday ‘𝑥) +no ( bday ‘𝑦𝑂)) ⊆ ( bday ‘(𝑥 +s 𝑦𝑂)))) → (( bday ‘𝑥) +no ( bday ‘𝑦)) ⊆ ( bday ‘(𝑥 +s 𝑦))))
163	8, 13, 16, 20, 25, 162	ons2ind 28446	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (( bday ‘𝐴) +no ( bday ‘𝐵)) ⊆ ( bday ‘(𝐴 +s 𝐵)))
164	4, 163	eqssd 3955	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ( bday ‘(𝐴 +s 𝐵)) = (( bday ‘𝐴) +no ( bday ‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416   ⊆ wss 3906  ∩ cint 4913   class class class wbr 5110   E cep 5562  ^◡ccnv 5662   ↾ cres 5665  Ord word 6361  Oncon0 6362  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539  (class class class)co 7412   +no cnadd 8652   No csur 27782   <s clts 27783   bday cbday 27784   +_s cadds 28130  On_scons 28422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  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 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-1o 8454  df-2o 8455  df-nadd 8653  df-no 27785  df-lts 27786  df-bday 27787  df-les 27887  df-slts 27929  df-cuts 27931  df-0s 27978  df-made 27998  df-old 27999  df-left 28001  df-right 28002  df-norec2 28120  df-adds 28131  df-ons 28423

This theorem is referenced by:  bdaypw2bnd  28636  z12bdaylem2  28642