Theorem oncutlt 28494

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 28485	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
2		ltonsex 28492	. . . . 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 5416	. . . 4 ⊢ (𝐴 ∈ Ons → {𝐴} ∈ V)
5		ssrab2 4037	. . . . . 6 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ Ons
6		onssno 28484	. . . . . 6 ⊢ Ons ⊆ No
7	5, 6	sstri 3949	. . . . 5 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ No
8	7	a1i 11	. . . 4 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ No )
9	1	snssd 4757	. . . 4 ⊢ (𝐴 ∈ Ons → {𝐴} ⊆ No )
10		breq1 5117	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 <s 𝐴 ↔ 𝑦 <s 𝐴))
11	10	elrab 3653	. . . . . . . 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 4610	. . . . . . . 8 ⊢ (𝑧 ∈ {𝐴} ↔ 𝑧 = 𝐴)
14		breq2 5118	. . . . . . . 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 27998	. . 3 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝐴})
20		snelpwi 5430	. . . 4 ⊢ (𝐴 ∈ No → {𝐴} ∈ 𝒫 No )
21		nulsgts 28006	. . . 4 ⊢ ({𝐴} ∈ 𝒫 No → {𝐴} <<s ∅)
22	1, 20, 21	3syl 19	. . 3 ⊢ (𝐴 ∈ Ons → {𝐴} <<s ∅)
23		sltssep 27997	. . . . . . 7 ⊢ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} → ∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}∀𝑤 ∈ {𝑦}𝑧 <s 𝑤)
24		vex 3462	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
25		breq2 5118	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑧 <s 𝑤 ↔ 𝑧 <s 𝑦))
26	24, 25	ralsn 4652	. . . . . . . . 9 ⊢ (∀𝑤 ∈ {𝑦}𝑧 <s 𝑤 ↔ 𝑧 <s 𝑦)
27	26	ralbii 3114	. . . . . . . 8 ⊢ (∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}∀𝑤 ∈ {𝑦}𝑧 <s 𝑤 ↔ ∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}𝑧 <s 𝑦)
28		breq1 5117	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 <s 𝐴 ↔ 𝑧 <s 𝐴))
29	28	ralrab 3660	. . . . . . . 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 6901	. . . . . . . . . . . . 13 ⊢ ( L ‘𝑦) ∈ V
33		fvex 6901	. . . . . . . . . . . . 13 ⊢ ( R ‘𝑦) ∈ V
34	32, 33	unex 7755	. . . . . . . . . . . 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 28103	. . . . . . . . . . . . 13 ⊢ ( L ‘𝑦) ⊆ No
37		rightssno 28104	. . . . . . . . . . . . 13 ⊢ ( R ‘𝑦) ⊆ No
38	36, 37	unssi 4147	. . . . . . . . . . . 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 2767	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) = ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅))
41	35, 39, 40	elons2d 28489	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) ∈ Ons)
42	34	elpw 4571	. . . . . . . . . . . . . . . . 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 28006	. . . . . . . . . . . . . . . 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 4354	. . . . . . . . . . . . . . . . . 18 ⊢ ((( L ‘𝑦) ∪ ( R ‘𝑦)) ∪ ∅) = (( L ‘𝑦) ∪ ( R ‘𝑦))
47		lrold 28127	. . . . . . . . . . . . . . . . . 18 ⊢ (( L ‘𝑦) ∪ ( R ‘𝑦)) = ( O ‘( bday ‘𝑦))
48	46, 47	eqtri 2789	. . . . . . . . . . . . . . . . 17 ⊢ ((( L ‘𝑦) ∪ ( R ‘𝑦)) ∪ ∅) = ( O ‘( bday ‘𝑦))
49	48	imaeq2i 6065	. . . . . . . . . . . . . . . 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 27982	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( bday ‘𝑦) ∈ On
52		oldssno 28071	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( O ‘( bday ‘𝑦)) ⊆ No
53	52	sseli 3936	. . . . . . . . . . . . . . . . . . . . 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 28131	. . . . . . . . . . . . . . . . . . . 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 3160	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ∀𝑧 ∈ ( O ‘( bday ‘𝑦))( bday ‘𝑧) ∈ ( bday ‘𝑦))
59		bdayfun 27977	. . . . . . . . . . . . . . . . . 18 ⊢ Fun bday
60		bdaydm 27979	. . . . . . . . . . . . . . . . . . 19 ⊢ dom bday = No
61	52, 60	sseqtrri 3989	. . . . . . . . . . . . . . . . . 18 ⊢ ( O ‘( bday ‘𝑦)) ⊆ dom bday
62		funimass4 6952	. . . . . . . . . . . . . . . . . 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 3978	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ( bday “ ((( L ‘𝑦) ∪ ( R ‘𝑦)) ∪ ∅)) ⊆ ( bday ‘𝑦))
66		cutbdaybnd 28025	. . . . . . . . . . . . . . . 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 27982	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅)) ∈ On
71		bdayon 27982	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝐴) ∈ On
72		ontr2 6416	. . . . . . . . . . . . . . 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 28013	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) ∈ No )
76		oldbday 28131	. . . . . . . . . . . . . 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 28483	. . . . . . . . . . . . . . . 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 4125	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → (( L ‘𝐴) ∪ ( R ‘𝐴)) = (( L ‘𝐴) ∪ ∅))
83		lrold 28127	. . . . . . . . . . . . 13 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))
84		un0 4354	. . . . . . . . . . . . 13 ⊢ (( L ‘𝐴) ∪ ∅) = ( L ‘𝐴)
85	82, 83, 84	3eqtr3g 2824	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
86	78, 85	eleqtrd 2868	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) ∈ ( L ‘𝐴))
87		leftlt 28083	. . . . . . . . . . 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 27968	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ No → 𝑦 ≤s 𝑦)
91		lrcut 28134	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ No → (( L ‘𝑦) |s ( R ‘𝑦)) = 𝑦)
92	90, 91	breqtrrd 5144	. . . . . . . . . . . . . 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 4119	. . . . . . . . . . . . . . . 16 ⊢ (( R ‘𝑦) = ∅ → (( L ‘𝑦) ∪ ( R ‘𝑦)) = (( L ‘𝑦) ∪ ∅))
95		un0 4354	. . . . . . . . . . . . . . . 16 ⊢ (( L ‘𝑦) ∪ ∅) = ( L ‘𝑦)
96	94, 95	eqtrdi 2817	. . . . . . . . . . . . . . 15 ⊢ (( R ‘𝑦) = ∅ → (( L ‘𝑦) ∪ ( R ‘𝑦)) = ( L ‘𝑦))
97		eqcom 2773	. . . . . . . . . . . . . . . 16 ⊢ (( R ‘𝑦) = ∅ ↔ ∅ = ( R ‘𝑦))
98	97	biimpi 219	. . . . . . . . . . . . . . 15 ⊢ (( R ‘𝑦) = ∅ → ∅ = ( R ‘𝑦))
99	96, 98	oveq12d 7441	. . . . . . . . . . . . . 14 ⊢ (( R ‘𝑦) = ∅ → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) = (( L ‘𝑦) |s ( R ‘𝑦)))
100	99	breq2d 5126	. . . . . . . . . . . . 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 4310	. . . . . . . . . . . . . . . . . 18 ⊢ (( R ‘𝑦) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ ( R ‘𝑦))
105		breq2 5118	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑤 → (𝑦 ≤s 𝑧 ↔ 𝑦 ≤s 𝑤))
106		elun2 4139	. . . . . . . . . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . . . . . . . . . . . . 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 28084	. . . . . . . . . . . . . . . . . . . . . . 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 27974	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ( R ‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → 𝑦 ≤s 𝑤)
114	105, 107, 113	rspcedvdw 3587	. . . . . . . . . . . . . . . . . . . 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 1963	. . . . . . . . . . . . . . . . . 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 28092	. . . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . . 16 ⊢ ((( R ‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → 𝑦 = (( L ‘𝑦) |s ( R ‘𝑦)))
125		eqidd 2767	. . . . . . . . . . . . . . . 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 28032	. . . . . . . . . . . . . . 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 27974	. . . . . . . . . . . . 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 3044	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → 𝑦 ≤s ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅))
131		lenlts 27953	. . . . . . . . . . . 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 5117	. . . . . . . . . . . 12 ⊢ (𝑧 = ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) → (𝑧 <s 𝐴 ↔ ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) <s 𝐴))
135		breq1 5117	. . . . . . . . . . . . 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 3584	. . . . . . . . . 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 6402	. . . . . . . . . 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 3122	. . . . . . . 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 3160	. . 3 ⊢ (𝐴 ∈ Ons → ∀𝑦 ∈ No (({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} ∧ {𝑦} <<s ∅) → ( bday ‘𝐴) ⊆ ( bday ‘𝑦)))
150	3, 8	elpwd 4573	. . . . 5 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∈ 𝒫 No )
151		nulsgts 28006	. . . . 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 28016	. . . 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 2772	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 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2961  ∀wral 3082  ∃wrex 3092  {crab 3419  Vcvv 3458   ∪ cun 3906   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567  {csn 4594   class class class wbr 5114  dom cdm 5666   “ cima 5669  Oncon0 6367  Fun wfun 6537  ‘cfv 6543  (class class class)co 7423   No csur 27841   <s clts 27842   bday cbday 27843   ≤s cles 27945   <<s cslts 27987   |s ccuts 27989   O cold 28053   L cleft 28055   R cright 28056  On_scons 28481

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5243  ax-sep 5262  ax-nul 5274  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3928  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4878  df-int 4918  df-iun 4963  df-br 5115  df-opab 5179  df-mpt 5198  df-tr 5224  df-id 5561  df-eprel 5566  df-po 5574  df-so 5575  df-fr 5619  df-we 5621  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6309  df-ord 6370  df-on 6371  df-suc 6373  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7380  df-ov 7426  df-oprab 7427  df-mpo 7428  df-2nd 7996  df-frecs 8287  df-wrecs 8318  df-recs 8367  df-1o 8462  df-2o 8463  df-no 27844  df-lts 27845  df-bday 27846  df-les 27946  df-slts 27988  df-cuts 27990  df-made 28057  df-old 28058  df-new 28059  df-left 28060  df-right 28061  df-ons 28482

This theorem is used by:  bdayons  28506  onsfi  28586  n0cutlt  28589