Theorem addonbday 28670

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 28646	. . 3 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No)
2		onno 28646	. . 3 ⊢ (𝐵 ∈ Ons → 𝐵 ∈ No)
3		addbday 28409	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (bday‘(𝐴 +s 𝐵)) ⊆ ((bday‘𝐴) +no (bday‘𝐵)))
4	1, 2, 3	syl2an 608	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (bday‘(𝐴 +s 𝐵)) ⊆ ((bday‘𝐴) +no (bday‘𝐵)))
5		fveq2 6885	. . . . 5 ⊢ (𝑥 = 𝑥𝑂 → (bday‘𝑥) = (bday‘𝑥𝑂))
6	5	oveq1d 7435	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → ((bday‘𝑥) +no (bday‘𝑦)) = ((bday‘𝑥𝑂) +no (bday‘𝑦)))
7		fvoveq1 7443	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (bday‘(𝑥 +s 𝑦)) = (bday‘(𝑥𝑂 +s 𝑦)))
8	6, 7	sseq12d 3941	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (((bday‘𝑥) +no (bday‘𝑦)) ⊆ (bday‘(𝑥 +s 𝑦)) ↔ ((bday‘𝑥𝑂) +no (bday‘𝑦)) ⊆ (bday‘(𝑥𝑂 +s 𝑦))))
9		fveq2 6885	. . . . 5 ⊢ (𝑦 = 𝑦𝑂 → (bday‘𝑦) = (bday‘𝑦𝑂))
10	9	oveq2d 7436	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → ((bday‘𝑥𝑂) +no (bday‘𝑦)) = ((bday‘𝑥𝑂) +no (bday‘𝑦𝑂)))
11		oveq2 7428	. . . . 5 ⊢ (𝑦 = 𝑦𝑂 → (𝑥𝑂 +s 𝑦) = (𝑥𝑂 +s 𝑦𝑂))
12	11	fveq2d 6889	. . . 4 ⊢ (𝑦 = 𝑦𝑂 → (bday‘(𝑥𝑂 +s 𝑦)) = (bday‘(𝑥𝑂 +s 𝑦𝑂)))
13	10, 12	sseq12d 3941	. . 3 ⊢ (𝑦 = 𝑦𝑂 → (((bday‘𝑥𝑂) +no (bday‘𝑦)) ⊆ (bday‘(𝑥𝑂 +s 𝑦)) ↔ ((bday‘𝑥𝑂) +no (bday‘𝑦𝑂)) ⊆ (bday‘(𝑥𝑂 +s 𝑦𝑂))))
14	5	oveq1d 7435	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → ((bday‘𝑥) +no (bday‘𝑦𝑂)) = ((bday‘𝑥𝑂) +no (bday‘𝑦𝑂)))
15		fvoveq1 7443	. . . 4 ⊢ (𝑥 = 𝑥𝑂 → (bday‘(𝑥 +s 𝑦𝑂)) = (bday‘(𝑥𝑂 +s 𝑦𝑂)))
16	14, 15	sseq12d 3941	. . 3 ⊢ (𝑥 = 𝑥𝑂 → (((bday‘𝑥) +no (bday‘𝑦𝑂)) ⊆ (bday‘(𝑥 +s 𝑦𝑂)) ↔ ((bday‘𝑥𝑂) +no (bday‘𝑦𝑂)) ⊆ (bday‘(𝑥𝑂 +s 𝑦𝑂))))
17		fveq2 6885	. . . . 5 ⊢ (𝑥 = 𝐴 → (bday‘𝑥) = (bday‘𝐴))
18	17	oveq1d 7435	. . . 4 ⊢ (𝑥 = 𝐴 → ((bday‘𝑥) +no (bday‘𝑦)) = ((bday‘𝐴) +no (bday‘𝑦)))
19		fvoveq1 7443	. . . 4 ⊢ (𝑥 = 𝐴 → (bday‘(𝑥 +s 𝑦)) = (bday‘(𝐴 +s 𝑦)))
20	18, 19	sseq12d 3941	. . 3 ⊢ (𝑥 = 𝐴 → (((bday‘𝑥) +no (bday‘𝑦)) ⊆ (bday‘(𝑥 +s 𝑦)) ↔ ((bday‘𝐴) +no (bday‘𝑦)) ⊆ (bday‘(𝐴 +s 𝑦))))
21		fveq2 6885	. . . . 5 ⊢ (𝑦 = 𝐵 → (bday‘𝑦) = (bday‘𝐵))
22	21	oveq2d 7436	. . . 4 ⊢ (𝑦 = 𝐵 → ((bday‘𝐴) +no (bday‘𝑦)) = ((bday‘𝐴) +no (bday‘𝐵)))
23		oveq2 7428	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝐴 +s 𝑦) = (𝐴 +s 𝐵))
24	23	fveq2d 6889	. . . 4 ⊢ (𝑦 = 𝐵 → (bday‘(𝐴 +s 𝑦)) = (bday‘(𝐴 +s 𝐵)))
25	22, 24	sseq12d 3941	. . 3 ⊢ (𝑦 = 𝐵 → (((bday‘𝐴) +no (bday‘𝑦)) ⊆ (bday‘(𝐴 +s 𝑦)) ↔ ((bday‘𝐴) +no (bday‘𝐵)) ⊆ (bday‘(𝐴 +s 𝐵))))
26		bdayon 28143	. . . . . 6 ⊢ (bday‘𝑥) ∈ On
27		bdayon 28143	. . . . . 6 ⊢ (bday‘𝑦) ∈ On
28		naddov2 8691	. . . . . 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 705	. . . . 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 5105	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦𝑂 = (◡(bday ↾ Ons)‘𝑞) → (𝑦𝑂 <s 𝑦 ↔ (◡(bday ↾ Ons)‘𝑞) <s 𝑦))
32		fveq2 6885	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦𝑂 = (◡(bday ↾ Ons)‘𝑞) → (bday‘𝑦𝑂) = (bday‘(◡(bday ↾ Ons)‘𝑞)))
33	32	oveq2d 7436	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦𝑂 = (◡(bday ↾ Ons)‘𝑞) → ((bday‘𝑥) +no (bday‘𝑦𝑂)) = ((bday‘𝑥) +no (bday‘(◡(bday ↾ Ons)‘𝑞))))
34		oveq2 7428	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦𝑂 = (◡(bday ↾ Ons)‘𝑞) → (𝑥 +s 𝑦𝑂) = (𝑥 +s (◡(bday ↾ Ons)‘𝑞)))
35	34	fveq2d 6889	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦𝑂 = (◡(bday ↾ Ons)‘𝑞) → (bday‘(𝑥 +s 𝑦𝑂)) = (bday‘(𝑥 +s (◡(bday ↾ Ons)‘𝑞))))
36	33, 35	sseq12d 3941	. . . . . . . . . . . . . . . . . 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 28662	. . . . . . . . . . . . . . . . . . . 20 ⊢ (bday ↾ Ons) Isom <s , E (Ons, On)
40		isof1o 7331	. . . . . . . . . . . . . . . . . . . 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 7293	. . . . . . . . . . . . . . . . . . 19 ⊢ (((bday ↾ Ons):Ons–1-1-onto→On ∧ 𝑞 ∈ On) → (◡(bday ↾ Ons)‘𝑞) ∈ Ons)
43	41, 42	mpan 703	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑞 ∈ On → (◡(bday ↾ Ons)‘𝑞) ∈ Ons)
44	43	adantl 487	. . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . 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 28646	. . . . . . . . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . . . . . . . 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 28646	. . . . . . . . . . . . . . . . . . . 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 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) → 𝑥 ∈ Ons)
53		onno 28646	. . . . . . . . . . . . . . . . . . . 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 28388	. . . . . . . . . . . . . . . . . 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 28668	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ Ons ∧ (◡(bday ↾ Ons)‘𝑞) ∈ Ons) → (𝑥 +s (◡(bday ↾ Ons)‘𝑞)) ∈ Ons)
57	52, 44, 56	syl2anc 596	. . . . . . . . . . . . . . . . . . 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 28668	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (𝑥 +s 𝑦) ∈ Ons)
59	58	ad2antrr 739	. . . . . . . . . . . . . . . . . . 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 28658	. . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . . 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 28143	. . . . . . . . . . . . . . . . . 18 ⊢ (bday‘(◡(bday ↾ Ons)‘𝑞)) ∈ On
66		naddcl 8689	. . . . . . . . . . . . . . . . . 18 ⊢ (((bday‘𝑥) ∈ On ∧ (bday‘(◡(bday ↾ Ons)‘𝑞)) ∈ On) → ((bday‘𝑥) +no (bday‘(◡(bday ↾ Ons)‘𝑞))) ∈ On)
67	26, 65, 66	mp2an 705	. . . . . . . . . . . . . . . . 17 ⊢ ((bday‘𝑥) +no (bday‘(◡(bday ↾ Ons)‘𝑞))) ∈ On
68	67	onordi 6476	. . . . . . . . . . . . . . . 16 ⊢ Ord ((bday‘𝑥) +no (bday‘(◡(bday ↾ Ons)‘𝑞)))
69		bdayon 28143	. . . . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . 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 462	. . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∈ On → ((bday ↾ Ons)‘(◡(bday ↾ Ons)‘𝑞)) = (bday‘(◡(bday ↾ Ons)‘𝑞)))
76	75	adantl 487	. . . . . . . . . . . . . . 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 7436	. . . . . . . . . . . . . 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 2845	. . . . . . . . . . . . 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 28658	. . . . . . . . . . . . . 14 ⊢ (((◡(bday ↾ Ons)‘𝑞) ∈ Ons ∧ 𝑦 ∈ Ons) → ((◡(bday ↾ Ons)‘𝑞) <s 𝑦 ↔ (bday‘(◡(bday ↾ Ons)‘𝑞)) ∈ (bday‘𝑦)))
81	44, 49, 80	syl2anc 596	. . . . . . . . . . . . 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 7285	. . . . . . . . . . . . . . . . 17 ⊢ (((bday ↾ Ons):Ons–1-1-onto→On ∧ 𝑞 ∈ On) → ((bday ↾ Ons)‘(◡(bday ↾ Ons)‘𝑞)) = 𝑞)
83	41, 82	mpan 703	. . . . . . . . . . . . . . . 16 ⊢ (𝑞 ∈ On → ((bday ↾ Ons)‘(◡(bday ↾ Ons)‘𝑞)) = 𝑞)
84	75, 83	eqtr3d 2797	. . . . . . . . . . . . . . 15 ⊢ (𝑞 ∈ On → (bday‘(◡(bday ↾ Ons)‘𝑞)) = 𝑞)
85	84	eleq1d 2845	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ On → ((bday‘(◡(bday ↾ Ons)‘𝑞)) ∈ (bday‘𝑦) ↔ 𝑞 ∈ (bday‘𝑦)))
86	85	adantl 487	. . . . . . . . . . . . 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 7436	. . . . . . . . . . . . . 14 ⊢ (𝑞 ∈ On → ((bday‘𝑥) +no ((bday ↾ Ons)‘(◡(bday ↾ Ons)‘𝑞))) = ((bday‘𝑥) +no 𝑞))
89	88	eleq1d 2845	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 418	. . . . . . . . . 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 3153	. . . . . . 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 5105	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥𝑂 = (◡(bday ↾ Ons)‘𝑝) → (𝑥𝑂 <s 𝑥 ↔ (◡(bday ↾ Ons)‘𝑝) <s 𝑥))
98		fveq2 6885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥𝑂 = (◡(bday ↾ Ons)‘𝑝) → (bday‘𝑥𝑂) = (bday‘(◡(bday ↾ Ons)‘𝑝)))
99	98	oveq1d 7435	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = (◡(bday ↾ Ons)‘𝑝) → ((bday‘𝑥𝑂) +no (bday‘𝑦)) = ((bday‘(◡(bday ↾ Ons)‘𝑝)) +no (bday‘𝑦)))
100		fvoveq1 7443	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥𝑂 = (◡(bday ↾ Ons)‘𝑝) → (bday‘(𝑥𝑂 +s 𝑦)) = (bday‘((◡(bday ↾ Ons)‘𝑝) +s 𝑦)))
101	99, 100	sseq12d 3941	. . . . . . . . . . . . . . . . 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 7293	. . . . . . . . . . . . . . . . . 18 ⊢ (((bday ↾ Ons):Ons–1-1-onto→On ∧ 𝑝 ∈ On) → (◡(bday ↾ Ons)‘𝑝) ∈ Ons)
105	41, 104	mpan 703	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ On → (◡(bday ↾ Ons)‘𝑝) ∈ Ons)
106	105	adantl 487	. . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . 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 28646	. . . . . . . . . . . . . . . . . . 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 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) → 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 788	. . . . . . . . . . . . . . . . . . 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 28389	. . . . . . . . . . . . . . . . 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 28668	. . . . . . . . . . . . . . . . . . 19 ⊢ (((◡(bday ↾ Ons)‘𝑝) ∈ Ons ∧ 𝑦 ∈ Ons) → ((◡(bday ↾ Ons)‘𝑝) +s 𝑦) ∈ Ons)
117	106, 113, 116	syl2anc 596	. . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . 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 28658	. . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . 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 460	. . . . . . . . . . . . . 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 28143	. . . . . . . . . . . . . . . . 17 ⊢ (bday‘(◡(bday ↾ Ons)‘𝑝)) ∈ On
125		naddcl 8689	. . . . . . . . . . . . . . . . 17 ⊢ (((bday‘(◡(bday ↾ Ons)‘𝑝)) ∈ On ∧ (bday‘𝑦) ∈ On) → ((bday‘(◡(bday ↾ Ons)‘𝑝)) +no (bday‘𝑦)) ∈ On)
126	124, 27, 125	mp2an 705	. . . . . . . . . . . . . . . 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 705	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 462	. . . . . . . . . . . 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 28658	. . . . . . . . . . . . . . . 16 ⊢ (((◡(bday ↾ Ons)‘𝑝) ∈ Ons ∧ 𝑥 ∈ Ons) → ((◡(bday ↾ Ons)‘𝑝) <s 𝑥 ↔ (bday‘(◡(bday ↾ Ons)‘𝑝)) ∈ (bday‘𝑥)))
133	105, 132	sylan 592	. . . . . . . . . . . . . . 15 ⊢ ((𝑝 ∈ On ∧ 𝑥 ∈ Ons) → ((◡(bday ↾ Ons)‘𝑝) <s 𝑥 ↔ (bday‘(◡(bday ↾ Ons)‘𝑝)) ∈ (bday‘𝑥)))
134	133	ancoms 464	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ Ons ∧ 𝑝 ∈ On) → ((◡(bday ↾ Ons)‘𝑝) <s 𝑥 ↔ (bday‘(◡(bday ↾ Ons)‘𝑝)) ∈ (bday‘𝑥)))
135	105	fvresd 6905	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ On → ((bday ↾ Ons)‘(◡(bday ↾ Ons)‘𝑝)) = (bday‘(◡(bday ↾ Ons)‘𝑝)))
136		f1ocnvfv2 7285	. . . . . . . . . . . . . . . . . 18 ⊢ (((bday ↾ Ons):Ons–1-1-onto→On ∧ 𝑝 ∈ On) → ((bday ↾ Ons)‘(◡(bday ↾ Ons)‘𝑝)) = 𝑝)
137	41, 136	mpan 703	. . . . . . . . . . . . . . . . 17 ⊢ (𝑝 ∈ On → ((bday ↾ Ons)‘(◡(bday ↾ Ons)‘𝑝)) = 𝑝)
138	135, 137	eqtr3d 2797	. . . . . . . . . . . . . . . 16 ⊢ (𝑝 ∈ On → (bday‘(◡(bday ↾ Ons)‘𝑝)) = 𝑝)
139	138	eleq1d 2845	. . . . . . . . . . . . . . 15 ⊢ (𝑝 ∈ On → ((bday‘(◡(bday ↾ Ons)‘𝑝)) ∈ (bday‘𝑥) ↔ 𝑝 ∈ (bday‘𝑥)))
140	139	adantl 487	. . . . . . . . . . . . . 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 765	. . . . . . . . . . . 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 7435	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ On → ((bday‘(◡(bday ↾ Ons)‘𝑝)) +no (bday‘𝑦)) = (𝑝 +no (bday‘𝑦)))
144	143	eleq1d 2845	. . . . . . . . . . . . 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 487	. . . . . . . . . . . 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 418	. . . . . . . . . 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 3153	. . . . . . 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 2849	. . . . . . . . . . 11 ⊢ (𝑎 = (bday‘(𝑥 +s 𝑦)) → (((bday‘𝑥) +no 𝑞) ∈ 𝑎 ↔ ((bday‘𝑥) +no 𝑞) ∈ (bday‘(𝑥 +s 𝑦))))
152	151	ralbidv 3185	. . . . . . . . . 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 2849	. . . . . . . . . . 11 ⊢ (𝑎 = (bday‘(𝑥 +s 𝑦)) → ((𝑝 +no (bday‘𝑦)) ∈ 𝑎 ↔ (𝑝 +no (bday‘𝑦)) ∈ (bday‘(𝑥 +s 𝑦))))
154	153	ralbidv 3185	. . . . . . . . . 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 644	. . . . . . . . 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 3645	. . . . . . . 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 595	. . . . . 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 4922	. . . . . 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 3946	. . . 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 418	. . 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 28666	. 2 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → ((bday‘𝐴) +no (bday‘𝐵)) ⊆ (bday‘(𝐴 +s 𝐵)))
164	4, 163	eqssd 3925	1 ⊢ ((𝐴 ∈ Ons ∧ 𝐵 ∈ Ons) → (bday‘(𝐴 +s 𝐵)) = ((bday‘𝐴) +no (bday‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {crab 3412   ⊆ wss 3895  ∩ cint 4906   class class class wbr 5102   E cep 5549  ^◡ccnv 5649   ↾ cres 5652  Ord word 6361  Oncon0 6362  –1-1-onto→wf1o 6537  ‘cfv 6538   Isom wiso 6539  (class class class)co 7420   +no cnadd 8677  Nocsur 28002   <s clts 28003  bdaycbday 28004   +_s cadds 28350  On_scons 28642

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

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 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  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-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-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec2 28340  df-adds 28351  df-ons 28643

This theorem is used by:  bdaypw2bnd  28856  z12bdaylem2  28862