Theorem bdayons 28268

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 6840	. . 3 ⊢ (𝑎 = 𝑏 → ( bday ‘𝑎) = ( bday ‘𝑏))
2		breq2 5089	. . . . . 6 ⊢ (𝑎 = 𝑏 → (𝑥 <s 𝑎 ↔ 𝑥 <s 𝑏))
3	2	rabbidv 3396	. . . . 5 ⊢ (𝑎 = 𝑏 → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} = {𝑥 ∈ Ons ∣ 𝑥 <s 𝑏})
4		breq1 5088	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 <s 𝑏 ↔ 𝑦 <s 𝑏))
5	4	cbvrabv 3399	. . . . 5 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑏} = {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}
6	3, 5	eqtrdi 2787	. . . 4 ⊢ (𝑎 = 𝑏 → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} = {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏})
7	6	imaeq2d 6025	. . 3 ⊢ (𝑎 = 𝑏 → ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) = ( bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))
8	1, 7	eqeq12d 2752	. 2 ⊢ (𝑎 = 𝑏 → (( bday ‘𝑎) = ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ ( bday ‘𝑏) = ( bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏})))
9		fveq2 6840	. . 3 ⊢ (𝑎 = 𝐴 → ( bday ‘𝑎) = ( bday ‘𝐴))
10		breq2 5089	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑥 <s 𝑎 ↔ 𝑥 <s 𝐴))
11	10	rabbidv 3396	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} = {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴})
12	11	imaeq2d 6025	. . 3 ⊢ (𝑎 = 𝐴 → ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) = ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}))
13	9, 12	eqeq12d 2752	. 2 ⊢ (𝑎 = 𝐴 → (( bday ‘𝑎) = ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ ( bday ‘𝐴) = ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴})))
14		oncutlt 28256	. . . . . . 7 ⊢ (𝑎 ∈ Ons → 𝑎 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} |s ∅))
15	14	adantr 480	. . . . . 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 6844	. . . . 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 28247	. . . . . . . . . 10 ⊢ (𝑎 ∈ Ons → 𝑎 ∈ No )
18		ltonsex 28254	. . . . . . . . . 10 ⊢ (𝑎 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ V)
19	17, 18	syl 17	. . . . . . . . 9 ⊢ (𝑎 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ V)
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → ( bday ‘𝑏) = ( bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ V)
21		ssrab2 4020	. . . . . . . . . 10 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ Ons
22		onssno 28246	. . . . . . . . . 10 ⊢ Ons ⊆ No
23	21, 22	sstri 3931	. . . . . . . . 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 4547	. . . . . . 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 27768	. . . . . . 7 ⊢ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ 𝒫 No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} <<s ∅)
27	25, 26	syl 17	. . . . . 6 ⊢ ((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → ( bday ‘𝑏) = ( bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) → {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} <<s ∅)
28		bdayfn 27741	. . . . . . . . . . . . 13 ⊢ bday Fn No
29		fvelimab 6912	. . . . . . . . . . . . 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 693	. . . . . . . . . . . 12 ⊢ (𝑞 ∈ ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ ∃𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ( bday ‘𝑧) = 𝑞)
31		breq1 5088	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 <s 𝑎 ↔ 𝑧 <s 𝑎))
32	31	rexrab 3642	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ( bday ‘𝑧) = 𝑞 ↔ ∃𝑧 ∈ Ons (𝑧 <s 𝑎 ∧ ( bday ‘𝑧) = 𝑞))
33	30, 32	bitri 275	. . . . . . . . . . 11 ⊢ (𝑞 ∈ ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ ∃𝑧 ∈ Ons (𝑧 <s 𝑎 ∧ ( bday ‘𝑧) = 𝑞))
34		breq1 5088	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝑧 → (𝑏 <s 𝑎 ↔ 𝑧 <s 𝑎))
35		fveq2 6840	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = 𝑧 → ( bday ‘𝑏) = ( bday ‘𝑧))
36		breq2 5089	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑏 = 𝑧 → (𝑦 <s 𝑏 ↔ 𝑦 <s 𝑧))
37	36	rabbidv 3396	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑏 = 𝑧 → {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏} = {𝑦 ∈ Ons ∣ 𝑦 <s 𝑧})
38		breq1 5088	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑥 = 𝑦 → (𝑥 <s 𝑧 ↔ 𝑦 <s 𝑧))
39	38	cbvrabv 3399	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧} = {𝑦 ∈ Ons ∣ 𝑦 <s 𝑧}
40	37, 39	eqtr4di 2789	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑏 = 𝑧 → {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏} = {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧})
41	40	imaeq2d 6025	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = 𝑧 → ( bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}) = ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧}))
42	35, 41	eqeq12d 2752	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝑧 → (( bday ‘𝑏) = ( bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}) ↔ ( bday ‘𝑧) = ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑧})))
43	34, 42	imbi12d 344	. . . . . . . . . . . . . . . . . . . . 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 3561	. . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . 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 454	. . . . . . . . . . . . . . . . 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 778	. . . . . . . . . . . . . . . . . . . 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 28247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ Ons → 𝑥 ∈ No )
50	49	adantl 481	. . . . . . . . . . . . . . . . . . . . 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 777	. . . . . . . . . . . . . . . . . . . . . 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 28247	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 ∈ Ons → 𝑧 ∈ No )
53	51, 52	syl 17	. . . . . . . . . . . . . . . . . . . . 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 775	. . . . . . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . . . . 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 27711	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ No ∧ 𝑧 ∈ No ∧ 𝑎 ∈ No ) → ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑎) → 𝑥 <s 𝑎))
57	50, 53, 55, 56	syl3anc 1374	. . . . . . . . . . . . . . . . . . . 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 695	. . . . . . . . . . . . . . . . . . 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 4015	. . . . . . . . . . . . . . . . . 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 6067	. . . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . 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 3956	. . . . . . . . . . . . . . . 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 3920	. . . . . . . . . . . . . . 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 2825	. . . . . . . . . . . . . . . . 17 ⊢ (( bday ‘𝑧) = 𝑞 → (𝑝 ∈ ( bday ‘𝑧) ↔ 𝑝 ∈ 𝑞))
65	64	imbi1d 341	. . . . . . . . . . . . . . . 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 223	. . . . . . . . . . . . . . 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 247	. . . . . . . . . . . . . 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 456	. . . . . . . . . . . . 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 410	. . . . . . . . . . . 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 3138	. . . . . . . . . . 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 242	. . . . . . . . . 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 411	. . . . . . . . 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 1930	. . . . . . . 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 6036	. . . . . . . . . . 11 ⊢ ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ⊆ ran bday
75		bdayrn 27743	. . . . . . . . . . 11 ⊢ ran bday = On
76	74, 75	sseqtri 3970	. . . . . . . . . 10 ⊢ ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ⊆ On
77		dford5 7738	. . . . . . . . . 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 710	. . . . . . . . 9 ⊢ (Ord ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ↔ Tr ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}))
79		dftr2 5194	. . . . . . . . 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 275	. . . . . . . 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 234	. . . . . . 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 27740	. . . . . . . 8 ⊢ Fun bday
83		funimaexg 6585	. . . . . . . 8 ⊢ ((Fun bday ∧ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∈ V) → ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ∈ V)
84	82, 20, 83	sylancr 588	. . . . . . 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 6334	. . . . . . 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 584	. . . . . 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 4334	. . . . . . . . 9 ⊢ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∪ ∅) = {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}
88	87	imaeq2i 6023	. . . . . . . 8 ⊢ ( bday “ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∪ ∅)) = ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})
89	88	eqimssi 3982	. . . . . . 7 ⊢ ( bday “ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ∪ ∅)) ⊆ ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎})
90		cutbdaybnd 27787	. . . . . . 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 1453	. . . . . 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 585	. . . . 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 3956	. . . 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 484	. . . . . . . 8 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → ( bday ‘𝑏) = ( bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ 𝑧 ∈ Ons) → 𝑧 ∈ Ons)
95		simpll 767	. . . . . . . 8 ⊢ (((𝑎 ∈ Ons ∧ ∀𝑏 ∈ Ons (𝑏 <s 𝑎 → ( bday ‘𝑏) = ( bday “ {𝑦 ∈ Ons ∣ 𝑦 <s 𝑏}))) ∧ 𝑧 ∈ Ons) → 𝑎 ∈ Ons)
96		onlts 28259	. . . . . . . 8 ⊢ ((𝑧 ∈ Ons ∧ 𝑎 ∈ Ons) → (𝑧 <s 𝑎 ↔ ( bday ‘𝑧) ∈ ( bday ‘𝑎)))
97	94, 95, 96	syl2anc 585	. . . . . . 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 229	. . . . . 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 3129	. . . . 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 27742	. . . . . . . 8 ⊢ dom bday = No
101	23, 100	sseqtrri 3971	. . . . . . 7 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ⊆ dom bday
102		funimass4 6904	. . . . . . 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 693	. . . . . 6 ⊢ (( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ⊆ ( bday ‘𝑎) ↔ ∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ( bday ‘𝑧) ∈ ( bday ‘𝑎))
104	31	ralrab 3640	. . . . . 6 ⊢ (∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎} ( bday ‘𝑧) ∈ ( bday ‘𝑎) ↔ ∀𝑧 ∈ Ons (𝑧 <s 𝑎 → ( bday ‘𝑧) ∈ ( bday ‘𝑎)))
105	103, 104	bitri 275	. . . . 5 ⊢ (( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝑎}) ⊆ ( bday ‘𝑎) ↔ ∀𝑧 ∈ Ons (𝑧 <s 𝑎 → ( bday ‘𝑧) ∈ ( bday ‘𝑎)))
106	99, 105	sylibr 234	. . . 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 3939	. . 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 412	. 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 28266	1 ⊢ (𝐴 ∈ Ons → ( bday ‘𝐴) = ( bday “ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  {crab 3389  Vcvv 3429   ∪ cun 3887   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541   class class class wbr 5085  Tr wtr 5192  dom cdm 5631  ran crn 5632   “ cima 5634  Ord word 6322  Oncon0 6323  Fun wfun 6492   Fn wfn 6493  ‘cfv 6498  (class class class)co 7367   No csur 27603   <s clts 27604   bday cbday 27605   <<s cslts 27749   |s ccuts 27751  On_scons 28243

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-1o 8405  df-2o 8406  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-made 27819  df-old 27820  df-new 27821  df-left 27822  df-right 27823  df-ons 28244

This theorem is referenced by: (None)