Theorem oncutlt 28655

Description: A surreal ordinal is the simplest number greater than all previous surreal ordinals. Theorem 15 of [Conway] p. 28. (Contributed by Scott Fenton, 4-Nov-2025.)

Assertion

Ref	Expression
oncutlt	⊢ (𝐴 ∈ Ons → 𝐴 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} |s ∅))

Distinct variable group:   𝑥,𝐴

Proof of Theorem oncutlt

Dummy variables 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onno 28646	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No)
2		ltonsex 28653	. . . . 5 ⊢ (𝐴 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∈ V)
3	1, 2	syl 18	. . . 4 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∈ V)
4		snexg 5397	. . . 4 ⊢ (𝐴 ∈ Ons → {𝐴} ∈ V)
5		ssrab2 4005	. . . . . 6 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ Ons
6		onssno 28645	. . . . . 6 ⊢ Ons ⊆ No
7	5, 6	sstri 3917	. . . . 5 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ No
8	7	a1i 11	. . . 4 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ No)
9	1	snssd 4743	. . . 4 ⊢ (𝐴 ∈ Ons → {𝐴} ⊆ No)
10		breq1 5105	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 <s 𝐴 ↔ 𝑦 <s 𝐴))
11	10	elrab 3644	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ↔ (𝑦 ∈ Ons ∧ 𝑦 <s 𝐴))
12	11	simprbi 503	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} → 𝑦 <s 𝐴)
13		velsn 4596	. . . . . . . 8 ⊢ (𝑧 ∈ {𝐴} ↔ 𝑧 = 𝐴)
14		breq2 5106	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝐴))
15	13, 14	sylbi 220	. . . . . . 7 ⊢ (𝑧 ∈ {𝐴} → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝐴))
16	12, 15	syl5ibrcom 250	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} → (𝑧 ∈ {𝐴} → 𝑦 <s 𝑧))
17	16	imp 412	. . . . 5 ⊢ ((𝑦 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∧ 𝑧 ∈ {𝐴}) → 𝑦 <s 𝑧)
18	17	3adant1 1148	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝑦 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∧ 𝑧 ∈ {𝐴}) → 𝑦 <s 𝑧)
19	3, 4, 8, 9, 18	sltsd 28159	. . 3 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝐴})
20		snelpwi 5411	. . . 4 ⊢ (𝐴 ∈ No → {𝐴} ∈ 𝒫 No)
21		nulsgts 28167	. . . 4 ⊢ ({𝐴} ∈ 𝒫 No → {𝐴} <<s ∅)
22	1, 20, 21	3syl 19	. . 3 ⊢ (𝐴 ∈ Ons → {𝐴} <<s ∅)
23		sltssep 28158	. . . . . . 7 ⊢ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} → ∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}∀𝑤 ∈ {𝑦}𝑧 <s 𝑤)
24		vex 3454	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
25		breq2 5106	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑧 <s 𝑤 ↔ 𝑧 <s 𝑦))
26	24, 25	ralsn 4638	. . . . . . . . 9 ⊢ (∀𝑤 ∈ {𝑦}𝑧 <s 𝑤 ↔ 𝑧 <s 𝑦)
27	26	ralbii 3108	. . . . . . . 8 ⊢ (∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}∀𝑤 ∈ {𝑦}𝑧 <s 𝑤 ↔ ∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}𝑧 <s 𝑦)
28		breq1 5105	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 <s 𝐴 ↔ 𝑧 <s 𝐴))
29	28	ralrab 3651	. . . . . . . 8 ⊢ (∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}𝑧 <s 𝑦 ↔ ∀𝑧 ∈ Ons (𝑧 <s 𝐴 → 𝑧 <s 𝑦))
30	27, 29	bitri 278	. . . . . . 7 ⊢ (∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}∀𝑤 ∈ {𝑦}𝑧 <s 𝑤 ↔ ∀𝑧 ∈ Ons (𝑧 <s 𝐴 → 𝑧 <s 𝑦))
31	23, 30	sylib 221	. . . . . 6 ⊢ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} → ∀𝑧 ∈ Ons (𝑧 <s 𝐴 → 𝑧 <s 𝑦))
32		fvex 6898	. . . . . . . . . . . . 13 ⊢ (L‘𝑦) ∈ V
33		fvex 6898	. . . . . . . . . . . . 13 ⊢ (R‘𝑦) ∈ V
34	32, 33	unex 7761	. . . . . . . . . . . 12 ⊢ ((L‘𝑦) ∪ (R‘𝑦)) ∈ V
35	34	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → ((L‘𝑦) ∪ (R‘𝑦)) ∈ V)
36		leftssno 28264	. . . . . . . . . . . . 13 ⊢ (L‘𝑦) ⊆ No
37		rightssno 28265	. . . . . . . . . . . . 13 ⊢ (R‘𝑦) ⊆ No
38	36, 37	unssi 4189	. . . . . . . . . . . 12 ⊢ ((L‘𝑦) ∪ (R‘𝑦)) ⊆ No
39	38	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → ((L‘𝑦) ∪ (R‘𝑦)) ⊆ No)
40		eqidd 2761	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) = (((L‘𝑦) ∪ (R‘𝑦)) |s ∅))
41	35, 39, 40	elons2d 28650	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ∈ Ons)
42	34	elpw 4560	. . . . . . . . . . . . . . . . 17 ⊢ (((L‘𝑦) ∪ (R‘𝑦)) ∈ 𝒫 No ↔ ((L‘𝑦) ∪ (R‘𝑦)) ⊆ No)
43	38, 42	mpbir 234	. . . . . . . . . . . . . . . 16 ⊢ ((L‘𝑦) ∪ (R‘𝑦)) ∈ 𝒫 No
44		nulsgts 28167	. . . . . . . . . . . . . . . 16 ⊢ (((L‘𝑦) ∪ (R‘𝑦)) ∈ 𝒫 No → ((L‘𝑦) ∪ (R‘𝑦)) <<s ∅)
45	43, 44	mp1i 14	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → ((L‘𝑦) ∪ (R‘𝑦)) <<s ∅)
46		un0 4343	. . . . . . . . . . . . . . . . . 18 ⊢ (((L‘𝑦) ∪ (R‘𝑦)) ∪ ∅) = ((L‘𝑦) ∪ (R‘𝑦))
47		lrold 28288	. . . . . . . . . . . . . . . . . 18 ⊢ ((L‘𝑦) ∪ (R‘𝑦)) = (O‘(bday‘𝑦))
48	46, 47	eqtri 2783	. . . . . . . . . . . . . . . . 17 ⊢ (((L‘𝑦) ∪ (R‘𝑦)) ∪ ∅) = (O‘(bday‘𝑦))
49	48	imaeq2i 6049	. . . . . . . . . . . . . . . 16 ⊢ (bday “ (((L‘𝑦) ∪ (R‘𝑦)) ∪ ∅)) = (bday “ (O‘(bday‘𝑦)))
50		simpr 490	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) ∧ 𝑧 ∈ (O‘(bday‘𝑦))) → 𝑧 ∈ (O‘(bday‘𝑦)))
51		bdayon 28143	. . . . . . . . . . . . . . . . . . . 20 ⊢ (bday‘𝑦) ∈ On
52		oldssno 28232	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (O‘(bday‘𝑦)) ⊆ No
53	52	sseli 3904	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ (O‘(bday‘𝑦)) → 𝑧 ∈ No)
54	53	adantl 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) ∧ 𝑧 ∈ (O‘(bday‘𝑦))) → 𝑧 ∈ No)
55		oldbday 28292	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((bday‘𝑦) ∈ On ∧ 𝑧 ∈ No) → (𝑧 ∈ (O‘(bday‘𝑦)) ↔ (bday‘𝑧) ∈ (bday‘𝑦)))
56	51, 54, 55	sylancr 599	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) ∧ 𝑧 ∈ (O‘(bday‘𝑦))) → (𝑧 ∈ (O‘(bday‘𝑦)) ↔ (bday‘𝑧) ∈ (bday‘𝑦)))
57	50, 56	mpbid 235	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) ∧ 𝑧 ∈ (O‘(bday‘𝑦))) → (bday‘𝑧) ∈ (bday‘𝑦))
58	57	ralrimiva 3154	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → ∀𝑧 ∈ (O‘(bday‘𝑦))(bday‘𝑧) ∈ (bday‘𝑦))
59		bdayfun 28138	. . . . . . . . . . . . . . . . . 18 ⊢ Fun bday
60		bdaydm 28140	. . . . . . . . . . . . . . . . . . 19 ⊢ dom bday = No
61	52, 60	sseqtrri 3957	. . . . . . . . . . . . . . . . . 18 ⊢ (O‘(bday‘𝑦)) ⊆ dom bday
62		funimass4 6949	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun bday ∧ (O‘(bday‘𝑦)) ⊆ dom bday) → ((bday “ (O‘(bday‘𝑦))) ⊆ (bday‘𝑦) ↔ ∀𝑧 ∈ (O‘(bday‘𝑦))(bday‘𝑧) ∈ (bday‘𝑦)))
63	59, 61, 62	mp2an 705	. . . . . . . . . . . . . . . . 17 ⊢ ((bday “ (O‘(bday‘𝑦))) ⊆ (bday‘𝑦) ↔ ∀𝑧 ∈ (O‘(bday‘𝑦))(bday‘𝑧) ∈ (bday‘𝑦))
64	58, 63	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (bday “ (O‘(bday‘𝑦))) ⊆ (bday‘𝑦))
65	49, 64	eqsstrid 3946	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (bday “ (((L‘𝑦) ∪ (R‘𝑦)) ∪ ∅)) ⊆ (bday‘𝑦))
66		cutbdaybnd 28186	. . . . . . . . . . . . . . . 16 ⊢ ((((L‘𝑦) ∪ (R‘𝑦)) <<s ∅ ∧ (bday‘𝑦) ∈ On ∧ (bday “ (((L‘𝑦) ∪ (R‘𝑦)) ∪ ∅)) ⊆ (bday‘𝑦)) → (bday‘(((L‘𝑦) ∪ (R‘𝑦)) |s ∅)) ⊆ (bday‘𝑦))
67	51, 66	mp3an2 1478	. . . . . . . . . . . . . . 15 ⊢ ((((L‘𝑦) ∪ (R‘𝑦)) <<s ∅ ∧ (bday “ (((L‘𝑦) ∪ (R‘𝑦)) ∪ ∅)) ⊆ (bday‘𝑦)) → (bday‘(((L‘𝑦) ∪ (R‘𝑦)) |s ∅)) ⊆ (bday‘𝑦))
68	45, 65, 67	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (bday‘(((L‘𝑦) ∪ (R‘𝑦)) |s ∅)) ⊆ (bday‘𝑦))
69		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (bday‘𝑦) ∈ (bday‘𝐴))
70		bdayon 28143	. . . . . . . . . . . . . . 15 ⊢ (bday‘(((L‘𝑦) ∪ (R‘𝑦)) |s ∅)) ∈ On
71		bdayon 28143	. . . . . . . . . . . . . . 15 ⊢ (bday‘𝐴) ∈ On
72		ontr2 6411	. . . . . . . . . . . . . . 15 ⊢ (((bday‘(((L‘𝑦) ∪ (R‘𝑦)) |s ∅)) ∈ On ∧ (bday‘𝐴) ∈ On) → (((bday‘(((L‘𝑦) ∪ (R‘𝑦)) |s ∅)) ⊆ (bday‘𝑦) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (bday‘(((L‘𝑦) ∪ (R‘𝑦)) |s ∅)) ∈ (bday‘𝐴)))
73	70, 71, 72	mp2an 705	. . . . . . . . . . . . . 14 ⊢ (((bday‘(((L‘𝑦) ∪ (R‘𝑦)) |s ∅)) ⊆ (bday‘𝑦) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (bday‘(((L‘𝑦) ∪ (R‘𝑦)) |s ∅)) ∈ (bday‘𝐴))
74	68, 69, 73	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (bday‘(((L‘𝑦) ∪ (R‘𝑦)) |s ∅)) ∈ (bday‘𝐴))
75	45	cutscld 28174	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ∈ No)
76		oldbday 28292	. . . . . . . . . . . . . 14 ⊢ (((bday‘𝐴) ∈ On ∧ (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ∈ No) → ((((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ∈ (O‘(bday‘𝐴)) ↔ (bday‘(((L‘𝑦) ∪ (R‘𝑦)) |s ∅)) ∈ (bday‘𝐴)))
77	71, 75, 76	sylancr 599	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → ((((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ∈ (O‘(bday‘𝐴)) ↔ (bday‘(((L‘𝑦) ∪ (R‘𝑦)) |s ∅)) ∈ (bday‘𝐴)))
78	74, 77	mpbird 260	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ∈ (O‘(bday‘𝐴)))
79		elons 28644	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ (R‘𝐴) = ∅))
80	79	simprbi 503	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Ons → (R‘𝐴) = ∅)
81	80	ad2antrr 739	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (R‘𝐴) = ∅)
82	81	uneq2d 4167	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → ((L‘𝐴) ∪ (R‘𝐴)) = ((L‘𝐴) ∪ ∅))
83		lrold 28288	. . . . . . . . . . . . 13 ⊢ ((L‘𝐴) ∪ (R‘𝐴)) = (O‘(bday‘𝐴))
84		un0 4343	. . . . . . . . . . . . 13 ⊢ ((L‘𝐴) ∪ ∅) = (L‘𝐴)
85	82, 83, 84	3eqtr3g 2818	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (O‘(bday‘𝐴)) = (L‘𝐴))
86	78, 85	eleqtrd 2862	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ∈ (L‘𝐴))
87		leftlt 28244	. . . . . . . . . . 11 ⊢ ((((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ∈ (L‘𝐴) → (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) <s 𝐴)
88	86, 87	syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) <s 𝐴)
89		simplr 781	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → 𝑦 ∈ No)
90		lesid 28129	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ No → 𝑦 ≤s 𝑦)
91		lrcut 28295	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ No → ((L‘𝑦) |s (R‘𝑦)) = 𝑦)
92	90, 91	breqtrrd 5132	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ No → 𝑦 ≤s ((L‘𝑦) |s (R‘𝑦)))
93	89, 92	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → 𝑦 ≤s ((L‘𝑦) |s (R‘𝑦)))
94		uneq2 4161	. . . . . . . . . . . . . . . 16 ⊢ ((R‘𝑦) = ∅ → ((L‘𝑦) ∪ (R‘𝑦)) = ((L‘𝑦) ∪ ∅))
95		un0 4343	. . . . . . . . . . . . . . . 16 ⊢ ((L‘𝑦) ∪ ∅) = (L‘𝑦)
96	94, 95	eqtrdi 2811	. . . . . . . . . . . . . . 15 ⊢ ((R‘𝑦) = ∅ → ((L‘𝑦) ∪ (R‘𝑦)) = (L‘𝑦))
97		eqcom 2767	. . . . . . . . . . . . . . . 16 ⊢ ((R‘𝑦) = ∅ ↔ ∅ = (R‘𝑦))
98	97	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ ((R‘𝑦) = ∅ → ∅ = (R‘𝑦))
99	96, 98	oveq12d 7438	. . . . . . . . . . . . . 14 ⊢ ((R‘𝑦) = ∅ → (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) = ((L‘𝑦) |s (R‘𝑦)))
100	99	breq2d 5114	. . . . . . . . . . . . 13 ⊢ ((R‘𝑦) = ∅ → (𝑦 ≤s (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ↔ 𝑦 ≤s ((L‘𝑦) |s (R‘𝑦))))
101	93, 100	imbitrrid 249	. . . . . . . . . . . 12 ⊢ ((R‘𝑦) = ∅ → (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → 𝑦 ≤s (((L‘𝑦) ∪ (R‘𝑦)) |s ∅)))
102		simprlr 792	. . . . . . . . . . . . . 14 ⊢ (((R‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → 𝑦 ∈ No)
103	75	adantl 487	. . . . . . . . . . . . . 14 ⊢ (((R‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ∈ No)
104		n0 4299	. . . . . . . . . . . . . . . . . 18 ⊢ ((R‘𝑦) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ (R‘𝑦))
105		breq2 5106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑤 → (𝑦 ≤s 𝑧 ↔ 𝑦 ≤s 𝑤))
106		elun2 4181	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ (R‘𝑦) → 𝑤 ∈ ((L‘𝑦) ∪ (R‘𝑦)))
107	106	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ (R‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → 𝑤 ∈ ((L‘𝑦) ∪ (R‘𝑦)))
108		simprlr 792	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ (R‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → 𝑦 ∈ No)
109	37	sseli 3904	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ (R‘𝑦) → 𝑤 ∈ No)
110	109	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ (R‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → 𝑤 ∈ No)
111		rightgt 28245	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ (R‘𝑦) → 𝑦 <s 𝑤)
112	111	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ (R‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → 𝑦 <s 𝑤)
113	108, 110, 112	ltlesd 28135	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ (R‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → 𝑦 ≤s 𝑤)
114	105, 107, 113	rspcedvdw 3579	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ (R‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → ∃𝑧 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝑦 ≤s 𝑧)
115	114	ex 418	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ (R‘𝑦) → (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → ∃𝑧 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝑦 ≤s 𝑧))
116	115	exlimiv 1962	. . . . . . . . . . . . . . . . . 18 ⊢ (∃𝑤 𝑤 ∈ (R‘𝑦) → (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → ∃𝑧 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝑦 ≤s 𝑧))
117	104, 116	sylbi 220	. . . . . . . . . . . . . . . . 17 ⊢ ((R‘𝑦) ≠ ∅ → (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → ∃𝑧 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝑦 ≤s 𝑧))
118	117	imp 412	. . . . . . . . . . . . . . . 16 ⊢ (((R‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → ∃𝑧 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝑦 ≤s 𝑧)
119	118	orcd 887	. . . . . . . . . . . . . . 15 ⊢ (((R‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → (∃𝑧 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝑦 ≤s 𝑧 ∨ ∃𝑤 ∈ (R‘𝑦)𝑤 ≤s (((L‘𝑦) ∪ (R‘𝑦)) |s ∅)))
120		lltr 28253	. . . . . . . . . . . . . . . . 17 ⊢ (L‘𝑦) <<s (R‘𝑦)
121	120	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((R‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → (L‘𝑦) <<s (R‘𝑦))
122	43, 44	mp1i 14	. . . . . . . . . . . . . . . 16 ⊢ (((R‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → ((L‘𝑦) ∪ (R‘𝑦)) <<s ∅)
123	102, 91	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((R‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → ((L‘𝑦) |s (R‘𝑦)) = 𝑦)
124	123	eqcomd 2766	. . . . . . . . . . . . . . . 16 ⊢ (((R‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → 𝑦 = ((L‘𝑦) |s (R‘𝑦)))
125		eqidd 2761	. . . . . . . . . . . . . . . 16 ⊢ (((R‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) = (((L‘𝑦) ∪ (R‘𝑦)) |s ∅))
126	121, 122, 124, 125	ltsrecd 28193	. . . . . . . . . . . . . . 15 ⊢ (((R‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → (𝑦 <s (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ↔ (∃𝑧 ∈ ((L‘𝑦) ∪ (R‘𝑦))𝑦 ≤s 𝑧 ∨ ∃𝑤 ∈ (R‘𝑦)𝑤 ≤s (((L‘𝑦) ∪ (R‘𝑦)) |s ∅))))
127	119, 126	mpbird 260	. . . . . . . . . . . . . 14 ⊢ (((R‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → 𝑦 <s (((L‘𝑦) ∪ (R‘𝑦)) |s ∅))
128	102, 103, 127	ltlesd 28135	. . . . . . . . . . . . 13 ⊢ (((R‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴))) → 𝑦 ≤s (((L‘𝑦) ∪ (R‘𝑦)) |s ∅))
129	128	ex 418	. . . . . . . . . . . 12 ⊢ ((R‘𝑦) ≠ ∅ → (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → 𝑦 ≤s (((L‘𝑦) ∪ (R‘𝑦)) |s ∅)))
130	101, 129	pm2.61ine 3038	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → 𝑦 ≤s (((L‘𝑦) ∪ (R‘𝑦)) |s ∅))
131		lenlts 28114	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ No ∧ (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ∈ No) → (𝑦 ≤s (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ↔ ¬ (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) <s 𝑦))
132	89, 75, 131	syl2anc 596	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → (𝑦 ≤s (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ↔ ¬ (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) <s 𝑦))
133	130, 132	mpbid 235	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → ¬ (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) <s 𝑦)
134		breq1 5105	. . . . . . . . . . . 12 ⊢ (𝑧 = (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) → (𝑧 <s 𝐴 ↔ (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) <s 𝐴))
135		breq1 5105	. . . . . . . . . . . . 13 ⊢ (𝑧 = (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) → (𝑧 <s 𝑦 ↔ (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) <s 𝑦))
136	135	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑧 = (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) → (¬ 𝑧 <s 𝑦 ↔ ¬ (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) <s 𝑦))
137	134, 136	anbi12d 644	. . . . . . . . . . 11 ⊢ (𝑧 = (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) → ((𝑧 <s 𝐴 ∧ ¬ 𝑧 <s 𝑦) ↔ ((((L‘𝑦) ∪ (R‘𝑦)) |s ∅) <s 𝐴 ∧ ¬ (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) <s 𝑦)))
138	137	rspcev 3576	. . . . . . . . . 10 ⊢ (((((L‘𝑦) ∪ (R‘𝑦)) |s ∅) ∈ Ons ∧ ((((L‘𝑦) ∪ (R‘𝑦)) |s ∅) <s 𝐴 ∧ ¬ (((L‘𝑦) ∪ (R‘𝑦)) |s ∅) <s 𝑦)) → ∃𝑧 ∈ Ons (𝑧 <s 𝐴 ∧ ¬ 𝑧 <s 𝑦))
139	41, 88, 133, 138	syl12anc 850	. . . . . . . . 9 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No) ∧ (bday‘𝑦) ∈ (bday‘𝐴)) → ∃𝑧 ∈ Ons (𝑧 <s 𝐴 ∧ ¬ 𝑧 <s 𝑦))
140	139	ex 418	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) → ((bday‘𝑦) ∈ (bday‘𝐴) → ∃𝑧 ∈ Ons (𝑧 <s 𝐴 ∧ ¬ 𝑧 <s 𝑦)))
141		ontri1 6397	. . . . . . . . . 10 ⊢ (((bday‘𝐴) ∈ On ∧ (bday‘𝑦) ∈ On) → ((bday‘𝐴) ⊆ (bday‘𝑦) ↔ ¬ (bday‘𝑦) ∈ (bday‘𝐴)))
142	71, 51, 141	mp2an 705	. . . . . . . . 9 ⊢ ((bday‘𝐴) ⊆ (bday‘𝑦) ↔ ¬ (bday‘𝑦) ∈ (bday‘𝐴))
143	142	con2bii 360	. . . . . . . 8 ⊢ ((bday‘𝑦) ∈ (bday‘𝐴) ↔ ¬ (bday‘𝐴) ⊆ (bday‘𝑦))
144		rexanali 3116	. . . . . . . 8 ⊢ (∃𝑧 ∈ Ons (𝑧 <s 𝐴 ∧ ¬ 𝑧 <s 𝑦) ↔ ¬ ∀𝑧 ∈ Ons (𝑧 <s 𝐴 → 𝑧 <s 𝑦))
145	140, 143, 144	3imtr3g 298	. . . . . . 7 ⊢ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) → (¬ (bday‘𝐴) ⊆ (bday‘𝑦) → ¬ ∀𝑧 ∈ Ons (𝑧 <s 𝐴 → 𝑧 <s 𝑦)))
146	145	con4d 116	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) → (∀𝑧 ∈ Ons (𝑧 <s 𝐴 → 𝑧 <s 𝑦) → (bday‘𝐴) ⊆ (bday‘𝑦)))
147	31, 146	syl5 35	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) → ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} → (bday‘𝐴) ⊆ (bday‘𝑦)))
148	147	adantrd 497	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No) → (({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} ∧ {𝑦} <<s ∅) → (bday‘𝐴) ⊆ (bday‘𝑦)))
149	148	ralrimiva 3154	. . 3 ⊢ (𝐴 ∈ Ons → ∀𝑦 ∈ No (({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} ∧ {𝑦} <<s ∅) → (bday‘𝐴) ⊆ (bday‘𝑦)))
150	3, 8	elpwd 4562	. . . . 5 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∈ 𝒫 No)
151		nulsgts 28167	. . . . 5 ⊢ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∈ 𝒫 No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s ∅)
152	150, 151	syl 18	. . . 4 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s ∅)
153		eqcuts2 28177	. . . 4 ⊢ (({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s ∅ ∧ 𝐴 ∈ No) → (({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} |s ∅) = 𝐴 ↔ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝐴} ∧ {𝐴} <<s ∅ ∧ ∀𝑦 ∈ No (({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} ∧ {𝑦} <<s ∅) → (bday‘𝐴) ⊆ (bday‘𝑦)))))
154	152, 1, 153	syl2anc 596	. . 3 ⊢ (𝐴 ∈ Ons → (({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} |s ∅) = 𝐴 ↔ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝐴} ∧ {𝐴} <<s ∅ ∧ ∀𝑦 ∈ No (({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} ∧ {𝑦} <<s ∅) → (bday‘𝐴) ⊆ (bday‘𝑦)))))
155	19, 22, 149, 154	mpbir3and 1361	. 2 ⊢ (𝐴 ∈ Ons → ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} |s ∅) = 𝐴)
156	155	eqcomd 2766	1 ⊢ (𝐴 ∈ Ons → 𝐴 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} |s ∅))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  𝒫 cpw 4556  {csn 4584   class class class wbr 5102  dom cdm 5650   “ cima 5653  Oncon0 6362  Fun wfun 6532  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   <s clts 28003  bdaycbday 28004   ≤s cles 28106   <<s cslts 28148   |s ccuts 28150  Ocold 28214  Lcleft 28216  Rcright 28217  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-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-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:  bdayons  28667  onsfi  28747  n0cutlt  28750