Theorem bdayons 28667

Description: The birthday of a surreal ordinal is the set of all previous ordinal birthdays. (Contributed by Scott Fenton, 7-Nov-2025.)

Assertion

Ref	Expression
bdayons	⊢ (𝐴 ∈ Ons → (bday‘𝐴) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}))

Distinct variable group:   𝑥,𝐴

Proof of Theorem bdayons

Dummy variables 𝑎 𝑏 𝑝 𝑞 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6885	. . 3 ⊢ (𝑎 = 𝑏 → (bday‘𝑎) = (bday‘𝑏))
2		breq2 5106	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑥 <s 𝑎 ↔ 𝑥 <s 𝑏))
3	2	rabbidv 3419	. . . . 5 ⊢ (𝑎 = 𝑏 → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} = {𝑥 ∈ Ons ∣ 𝑥 <s 𝑏})
4		breq1 5105	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 <s 𝑏 ↔ 𝑦 <s 𝑏))
5	4	cbvrabv 3422	. . . . 5 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑏} = {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}
6	3, 5	eqtrdi 2811	. . . 4 ⊢ (𝑎 = 𝑏 → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} = {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏})
7	6	imaeq2d 6051	. . 3 ⊢ (𝑎 = 𝑏 → (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))
8	1, 7	eqeq12d 2776	. 2 ⊢ (𝑎 = 𝑏 → ((bday‘𝑎) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏})))
9		fveq2 6885	. . 3 ⊢ (𝑎 = 𝐴 → (bday‘𝑎) = (bday‘𝐴))
10		breq2 5106	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑥 <s 𝑎 ↔ 𝑥 <s 𝐴))
11	10	rabbidv 3419	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} = {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴})
12	11	imaeq2d 6051	. . 3 ⊢ (𝑎 = 𝐴 → (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}))
13	9, 12	eqeq12d 2776	. 2 ⊢ (𝑎 = 𝐴 → ((bday‘𝑎) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ (bday‘𝐴) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴})))
14		oncutlt 28655	. . . . . . 7 ⊢ (𝑎 ∈ Ons → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
15	14	adantr 486	. . . . . 6 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
16	15	fveq2d 6889	. . . . 5 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → (bday‘𝑎) = (bday‘({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅)))
17		onno 28646	. . . . . . . . . 10 ⊢ (𝑎 ∈ Ons → 𝑎 ∈ No)
18		ltonsex 28653	. . . . . . . . . 10 ⊢ (𝑎 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ V)
19	17, 18	syl 18	. . . . . . . . 9 ⊢ (𝑎 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ V)
20	19	adantr 486	. . . . . . . 8 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ V)
21		ssrab2 4005	. . . . . . . . . 10 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ Ons
22		onssno 28645	. . . . . . . . . 10 ⊢ Ons ⊆ No
23	21, 22	sstri 3917	. . . . . . . . 9 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ No
24	23	a1i 11	. . . . . . . 8 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ No)
25	20, 24	elpwd 4562	. . . . . . 7 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ 𝒫 No)
26		nulsgts 28167	. . . . . . 7 ⊢ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ 𝒫 No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} <<s ∅)
27	25, 26	syl 18	. . . . . 6 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} <<s ∅)
28		bdayfn 28139	. . . . . . . . . . . . 13 ⊢ bday Fn No
29		fvelimab 6957	. . . . . . . . . . . . 13 ⊢ ((bday Fn No ∧ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ No) → (𝑞 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ ∃𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} (bday‘𝑧) = 𝑞))
30	28, 23, 29	mp2an 705	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ ∃𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} (bday‘𝑧) = 𝑞)
31		breq1 5105	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 <s 𝑎 ↔ 𝑧 <s 𝑎))
32	31	rexrab 3653	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} (bday‘𝑧) = 𝑞 ↔ ∃𝑧 ∈ Ons (𝑧 <s 𝑎 ∧ (bday‘𝑧) = 𝑞))
33	30, 32	bitri 278	. . . . . . . . . . 11 ⊢ (𝑞 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ ∃𝑧 ∈ Ons (𝑧 <s 𝑎 ∧ (bday‘𝑧) = 𝑞))
34		breq1 5105	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝑧 → (𝑏 <s 𝑎 ↔ 𝑧 <s 𝑎))
35		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = 𝑧 → (bday‘𝑏) = (bday‘𝑧))
36		breq2 5106	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 = 𝑧 → (𝑦 <s 𝑏 ↔ 𝑦 <s 𝑧))
37	36	rabbidv 3419	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = 𝑧 → {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏} = {𝑦 ∈ Ons ∣ 𝑦 <s 𝑧})
38		breq1 5105	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑦 → (𝑥 <s 𝑧 ↔ 𝑦 <s 𝑧))
39	38	cbvrabv 3422	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧} = {𝑦 ∈ Ons ∣ 𝑦 <s 𝑧}
40	37, 39	eqtr4di 2813	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = 𝑧 → {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏} = {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧})
41	40	imaeq2d 6051	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = 𝑧 → (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧}))
42	35, 41	eqeq12d 2776	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝑧 → ((bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}) ↔ (bday‘𝑧) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧})))
43	34, 42	imbi12d 347	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 = 𝑧 → ((𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏})) ↔ (𝑧 <s 𝑎 → (bday‘𝑧) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧}))))
44	43	rspccv 3573	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏})) → (𝑧 ∈ Ons → (𝑧 <s 𝑎 → (bday‘𝑧) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧}))))
45	44	imp 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏})) ∧ 𝑧 ∈ Ons) → (𝑧 <s 𝑎 → (bday‘𝑧) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧})))
46	45	adantll 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ 𝑧 ∈ Ons) → (𝑧 <s 𝑎 → (bday‘𝑧) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧})))
47	46	impr 460	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) → (bday‘𝑧) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧}))
48		simplrr 790	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) ∧ 𝑥 ∈ Ons) → 𝑧 <s 𝑎)
49		onno 28646	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ Ons → 𝑥 ∈ No)
50	49	adantl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) ∧ 𝑥 ∈ Ons) → 𝑥 ∈ No)
51		simplrl 789	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) ∧ 𝑥 ∈ Ons) → 𝑧 ∈ Ons)
52		onno 28646	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ Ons → 𝑧 ∈ No)
53	51, 52	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) ∧ 𝑥 ∈ Ons) → 𝑧 ∈ No)
54		simplll 787	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) ∧ 𝑥 ∈ Ons) → 𝑎 ∈ Ons)
55	54, 17	syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) ∧ 𝑥 ∈ Ons) → 𝑎 ∈ No)
56		ltstr 28109	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ No ∧ 𝑧 ∈ No ∧ 𝑎 ∈ No) → ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑎) → 𝑥 <s 𝑎))
57	50, 53, 55, 56	syl3anc 1398	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) ∧ 𝑥 ∈ Ons) → ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑎) → 𝑥 <s 𝑎))
58	48, 57	mpan2d 707	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) ∧ 𝑥 ∈ Ons) → (𝑥 <s 𝑧 → 𝑥 <s 𝑎))
59	58	ss2rabdv 4000	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧} ⊆ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})
60		imass2 6054	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑧} ⊆ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} → (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧}) ⊆ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))
61	59, 60	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) → (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧}) ⊆ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))
62	47, 61	eqsstrd 3942	. . . . . . . . . . . . . . . 16 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) → (bday‘𝑧) ⊆ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))
63	62	sseld 3907	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) → (𝑝 ∈ (bday‘𝑧) → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})))
64		eleq2 2849	. . . . . . . . . . . . . . . . 17 ⊢ ((bday‘𝑧) = 𝑞 → (𝑝 ∈ (bday‘𝑧) ↔ 𝑝 ∈ 𝑞))
65	64	imbi1d 344	. . . . . . . . . . . . . . . 16 ⊢ ((bday‘𝑧) = 𝑞 → ((𝑝 ∈ (bday‘𝑧) → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})) ↔ (𝑝 ∈ 𝑞 → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))))
66	65	bicomd 226	. . . . . . . . . . . . . . 15 ⊢ ((bday‘𝑧) = 𝑞 → ((𝑝 ∈ 𝑞 → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})) ↔ (𝑝 ∈ (bday‘𝑧) → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))))
67	63, 66	syl5ibrcom 250	. . . . . . . . . . . . . 14 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ (𝑧 ∈ Ons ∧ 𝑧 <s 𝑎)) → ((bday‘𝑧) = 𝑞 → (𝑝 ∈ 𝑞 → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))))
68	67	expr 462	. . . . . . . . . . . . 13 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ 𝑧 ∈ Ons) → (𝑧 <s 𝑎 → ((bday‘𝑧) = 𝑞 → (𝑝 ∈ 𝑞 → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})))))
69	68	impd 416	. . . . . . . . . . . 12 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ 𝑧 ∈ Ons) → ((𝑧 <s 𝑎 ∧ (bday‘𝑧) = 𝑞) → (𝑝 ∈ 𝑞 → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))))
70	69	rexlimdva 3163	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → (∃𝑧 ∈ Ons (𝑧 <s 𝑎 ∧ (bday‘𝑧) = 𝑞) → (𝑝 ∈ 𝑞 → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))))
71	33, 70	biimtrid 245	. . . . . . . . . 10 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → (𝑞 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) → (𝑝 ∈ 𝑞 → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))))
72	71	impcomd 417	. . . . . . . . 9 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → ((𝑝 ∈ 𝑞 ∧ 𝑞 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})) → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})))
73	72	alrimivv 1960	. . . . . . . 8 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → ∀𝑝∀𝑞((𝑝 ∈ 𝑞 ∧ 𝑞 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})) → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})))
74		imassrn 6197	. . . . . . . . . . 11 ⊢ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ⊆ ran bday
75		bdayrn 28142	. . . . . . . . . . 11 ⊢ ran bday = On
76	74, 75	sseqtri 3956	. . . . . . . . . 10 ⊢ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ⊆ On
77		dford5 7798	. . . . . . . . . 10 ⊢ (Ord (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ ((bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ⊆ On ∧ Tr (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})))
78	76, 77	mpbiran 722	. . . . . . . . 9 ⊢ (Ord (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ Tr (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))
79		dftr2 5213	. . . . . . . . 9 ⊢ (Tr (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ ∀𝑝∀𝑞((𝑝 ∈ 𝑞 ∧ 𝑞 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})) → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})))
80	78, 79	bitri 278	. . . . . . . 8 ⊢ (Ord (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ ∀𝑝∀𝑞((𝑝 ∈ 𝑞 ∧ 𝑞 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})) → 𝑝 ∈ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})))
81	73, 80	sylibr 237	. . . . . . 7 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → Ord (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))
82		bdayfun 28138	. . . . . . . 8 ⊢ Fun bday
83		funimaexg 6626	. . . . . . . 8 ⊢ ((Fun bday ∧ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ V) → (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ∈ V)
84	82, 20, 83	sylancr 599	. . . . . . 7 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ∈ V)
85		elon2 6373	. . . . . . 7 ⊢ ((bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ∈ On ↔ (Ord (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ∧ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ∈ V))
86	81, 84, 85	sylanbrc 595	. . . . . 6 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ∈ On)
87		un0 4343	. . . . . . . . 9 ⊢ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∪ ∅) = {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}
88	87	imaeq2i 6049	. . . . . . . 8 ⊢ (bday “ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∪ ∅)) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})
89	88	eqimssi 3968	. . . . . . 7 ⊢ (bday “ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∪ ∅)) ⊆ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})
90		cutbdaybnd 28186	. . . . . . 7 ⊢ (({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} <<s ∅ ∧ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ∈ On ∧ (bday “ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∪ ∅)) ⊆ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})) → (bday‘({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅)) ⊆ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))
91	89, 90	mp3an3 1479	. . . . . 6 ⊢ (({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} <<s ∅ ∧ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ∈ On) → (bday‘({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅)) ⊆ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))
92	27, 86, 91	syl2anc 596	. . . . 5 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → (bday‘({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅)) ⊆ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))
93	16, 92	eqsstrd 3942	. . . 4 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → (bday‘𝑎) ⊆ (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))
94		simpr 490	. . . . . . . 8 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ 𝑧 ∈ Ons) → 𝑧 ∈ Ons)
95		simpll 779	. . . . . . . 8 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ 𝑧 ∈ Ons) → 𝑎 ∈ Ons)
96		onlts 28658	. . . . . . . 8 ⊢ ((𝑧 ∈ Ons ∧ 𝑎 ∈ Ons) → (𝑧 <s 𝑎 ↔ (bday‘𝑧) ∈ (bday‘𝑎)))
97	94, 95, 96	syl2anc 596	. . . . . . 7 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ 𝑧 ∈ Ons) → (𝑧 <s 𝑎 ↔ (bday‘𝑧) ∈ (bday‘𝑎)))
98	97	biimpd 232	. . . . . 6 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ 𝑧 ∈ Ons) → (𝑧 <s 𝑎 → (bday‘𝑧) ∈ (bday‘𝑎)))
99	98	ralrimiva 3154	. . . . 5 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → ∀𝑧 ∈ Ons (𝑧 <s 𝑎 → (bday‘𝑧) ∈ (bday‘𝑎)))
100		bdaydm 28140	. . . . . . . 8 ⊢ dom bday = No
101	23, 100	sseqtrri 3957	. . . . . . 7 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ dom bday
102		funimass4 6949	. . . . . . 7 ⊢ ((Fun bday ∧ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ dom bday) → ((bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ⊆ (bday‘𝑎) ↔ ∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} (bday‘𝑧) ∈ (bday‘𝑎)))
103	82, 101, 102	mp2an 705	. . . . . 6 ⊢ ((bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ⊆ (bday‘𝑎) ↔ ∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} (bday‘𝑧) ∈ (bday‘𝑎))
104	31	ralrab 3651	. . . . . 6 ⊢ (∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} (bday‘𝑧) ∈ (bday‘𝑎) ↔ ∀𝑧 ∈ Ons (𝑧 <s 𝑎 → (bday‘𝑧) ∈ (bday‘𝑎)))
105	103, 104	bitri 278	. . . . 5 ⊢ ((bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ⊆ (bday‘𝑎) ↔ ∀𝑧 ∈ Ons (𝑧 <s 𝑎 → (bday‘𝑧) ∈ (bday‘𝑎)))
106	99, 105	sylibr 237	. . . 4 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ⊆ (bday‘𝑎))
107	93, 106	eqssd 3925	. . 3 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → (bday‘𝑎) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))
108	107	ex 418	. 2 ⊢ (𝑎 ∈ Ons → (∀𝑏 ∈ Ons (𝑏 <s 𝑎 → (bday‘𝑏) = (bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏})) → (bday‘𝑎) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})))
109	8, 13, 108	onsis 28665	1 ⊢ (𝐴 ∈ Ons → (bday‘𝐴) = (bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  𝒫 cpw 4556   class class class wbr 5102  Tr wtr 5211  dom cdm 5650  ran crn 5651   “ cima 5653  Ord word 6361  Oncon0 6362  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   <s clts 28003  bdaycbday 28004   <<s cslts 28148   |s ccuts 28150  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-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-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-made 28218  df-old 28219  df-new 28220  df-left 28221  df-right 28222  df-ons 28643

This theorem is used by: (None)