Theorem onscutlt 28201

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

Distinct variable group:   𝑥,𝐴

Proof of Theorem onscutlt

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

Step	Hyp	Ref	Expression
1		onsno 28192	. . . . 5 ⊢ (𝐴 ∈ Ons → 𝐴 ∈ No )
2		sltonex 28199	. . . . 5 ⊢ (𝐴 ∈ No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∈ V)
3	1, 2	syl 17	. . . 4 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∈ V)
4		snexg 5371	. . . 4 ⊢ (𝐴 ∈ Ons → {𝐴} ∈ V)
5		ssrab2 4027	. . . . . 6 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ Ons
6		onssno 28191	. . . . . 6 ⊢ Ons ⊆ No
7	5, 6	sstri 3939	. . . . 5 ⊢ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ No
8	7	a1i 11	. . . 4 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ⊆ No )
9	1	snssd 4758	. . . 4 ⊢ (𝐴 ∈ Ons → {𝐴} ⊆ No )
10		breq1 5092	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 <s 𝐴 ↔ 𝑦 <s 𝐴))
11	10	elrab 3642	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ↔ (𝑦 ∈ Ons ∧ 𝑦 <s 𝐴))
12	11	simprbi 496	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} → 𝑦 <s 𝐴)
13		velsn 4589	. . . . . . . 8 ⊢ (𝑧 ∈ {𝐴} ↔ 𝑧 = 𝐴)
14		breq2 5093	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝐴))
15	13, 14	sylbi 217	. . . . . . 7 ⊢ (𝑧 ∈ {𝐴} → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝐴))
16	12, 15	syl5ibrcom 247	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} → (𝑧 ∈ {𝐴} → 𝑦 <s 𝑧))
17	16	imp 406	. . . . 5 ⊢ ((𝑦 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∧ 𝑧 ∈ {𝐴}) → 𝑦 <s 𝑧)
18	17	3adant1 1130	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝑦 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∧ 𝑧 ∈ {𝐴}) → 𝑦 <s 𝑧)
19	3, 4, 8, 9, 18	ssltd 27731	. . 3 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝐴})
20		snelpwi 5383	. . . 4 ⊢ (𝐴 ∈ No → {𝐴} ∈ 𝒫 No )
21		nulssgt 27739	. . . 4 ⊢ ({𝐴} ∈ 𝒫 No → {𝐴} <<s ∅)
22	1, 20, 21	3syl 18	. . 3 ⊢ (𝐴 ∈ Ons → {𝐴} <<s ∅)
23		ssltsep 27730	. . . . . . 7 ⊢ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} → ∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}∀𝑤 ∈ {𝑦}𝑧 <s 𝑤)
24		vex 3440	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
25		breq2 5093	. . . . . . . . . 10 ⊢ (𝑤 = 𝑦 → (𝑧 <s 𝑤 ↔ 𝑧 <s 𝑦))
26	24, 25	ralsn 4631	. . . . . . . . 9 ⊢ (∀𝑤 ∈ {𝑦}𝑧 <s 𝑤 ↔ 𝑧 <s 𝑦)
27	26	ralbii 3078	. . . . . . . 8 ⊢ (∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}∀𝑤 ∈ {𝑦}𝑧 <s 𝑤 ↔ ∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}𝑧 <s 𝑦)
28		breq1 5092	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 <s 𝐴 ↔ 𝑧 <s 𝐴))
29	28	ralrab 3648	. . . . . . . 8 ⊢ (∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}𝑧 <s 𝑦 ↔ ∀𝑧 ∈ Ons (𝑧 <s 𝐴 → 𝑧 <s 𝑦))
30	27, 29	bitri 275	. . . . . . 7 ⊢ (∀𝑧 ∈ {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴}∀𝑤 ∈ {𝑦}𝑧 <s 𝑤 ↔ ∀𝑧 ∈ Ons (𝑧 <s 𝐴 → 𝑧 <s 𝑦))
31	23, 30	sylib 218	. . . . . 6 ⊢ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} → ∀𝑧 ∈ Ons (𝑧 <s 𝐴 → 𝑧 <s 𝑦))
32		fvex 6835	. . . . . . . . . . . . 13 ⊢ ( L ‘𝑦) ∈ V
33		fvex 6835	. . . . . . . . . . . . 13 ⊢ ( R ‘𝑦) ∈ V
34	32, 33	unex 7677	. . . . . . . . . . . 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 27826	. . . . . . . . . . . . 13 ⊢ ( L ‘𝑦) ⊆ No
37		rightssno 27827	. . . . . . . . . . . . 13 ⊢ ( R ‘𝑦) ⊆ No
38	36, 37	unssi 4138	. . . . . . . . . . . 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 2732	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) = ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅))
41	35, 39, 40	elons2d 28196	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) ∈ Ons)
42	34	elpw 4551	. . . . . . . . . . . . . . . . 17 ⊢ ((( L ‘𝑦) ∪ ( R ‘𝑦)) ∈ 𝒫 No ↔ (( L ‘𝑦) ∪ ( R ‘𝑦)) ⊆ No )
43	38, 42	mpbir 231	. . . . . . . . . . . . . . . 16 ⊢ (( L ‘𝑦) ∪ ( R ‘𝑦)) ∈ 𝒫 No
44		nulssgt 27739	. . . . . . . . . . . . . . . 16 ⊢ ((( L ‘𝑦) ∪ ( R ‘𝑦)) ∈ 𝒫 No → (( L ‘𝑦) ∪ ( R ‘𝑦)) <<s ∅)
45	43, 44	mp1i 13	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → (( L ‘𝑦) ∪ ( R ‘𝑦)) <<s ∅)
46		un0 4341	. . . . . . . . . . . . . . . . . 18 ⊢ ((( L ‘𝑦) ∪ ( R ‘𝑦)) ∪ ∅) = (( L ‘𝑦) ∪ ( R ‘𝑦))
47		lrold 27842	. . . . . . . . . . . . . . . . . 18 ⊢ (( L ‘𝑦) ∪ ( R ‘𝑦)) = ( O ‘( bday ‘𝑦))
48	46, 47	eqtri 2754	. . . . . . . . . . . . . . . . 17 ⊢ ((( L ‘𝑦) ∪ ( R ‘𝑦)) ∪ ∅) = ( O ‘( bday ‘𝑦))
49	48	imaeq2i 6006	. . . . . . . . . . . . . . . 16 ⊢ ( bday “ ((( L ‘𝑦) ∪ ( R ‘𝑦)) ∪ ∅)) = ( bday “ ( O ‘( bday ‘𝑦)))
50		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) ∧ 𝑧 ∈ ( O ‘( bday ‘𝑦))) → 𝑧 ∈ ( O ‘( bday ‘𝑦)))
51		bdayelon 27715	. . . . . . . . . . . . . . . . . . . 20 ⊢ ( bday ‘𝑦) ∈ On
52		oldssno 27802	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ( O ‘( bday ‘𝑦)) ⊆ No
53	52	sseli 3925	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 ∈ ( O ‘( bday ‘𝑦)) → 𝑧 ∈ No )
54	53	adantl 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) ∧ 𝑧 ∈ ( O ‘( bday ‘𝑦))) → 𝑧 ∈ No )
55		oldbday 27846	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((( bday ‘𝑦) ∈ On ∧ 𝑧 ∈ No ) → (𝑧 ∈ ( O ‘( bday ‘𝑦)) ↔ ( bday ‘𝑧) ∈ ( bday ‘𝑦)))
56	51, 54, 55	sylancr 587	. . . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) ∧ 𝑧 ∈ ( O ‘( bday ‘𝑦))) → ( bday ‘𝑧) ∈ ( bday ‘𝑦))
58	57	ralrimiva 3124	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ∀𝑧 ∈ ( O ‘( bday ‘𝑦))( bday ‘𝑧) ∈ ( bday ‘𝑦))
59		bdayfun 27711	. . . . . . . . . . . . . . . . . 18 ⊢ Fun bday
60		bdaydm 27713	. . . . . . . . . . . . . . . . . . 19 ⊢ dom bday = No
61	52, 60	sseqtrri 3979	. . . . . . . . . . . . . . . . . 18 ⊢ ( O ‘( bday ‘𝑦)) ⊆ dom bday
62		funimass4 6886	. . . . . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . . . 17 ⊢ (( bday “ ( O ‘( bday ‘𝑦))) ⊆ ( bday ‘𝑦) ↔ ∀𝑧 ∈ ( O ‘( bday ‘𝑦))( bday ‘𝑧) ∈ ( bday ‘𝑦))
64	58, 63	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ( bday “ ( O ‘( bday ‘𝑦))) ⊆ ( bday ‘𝑦))
65	49, 64	eqsstrid 3968	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ( bday “ ((( L ‘𝑦) ∪ ( R ‘𝑦)) ∪ ∅)) ⊆ ( bday ‘𝑦))
66		scutbdaybnd 27756	. . . . . . . . . . . . . . . 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 1451	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ( bday ‘((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅)) ⊆ ( bday ‘𝑦))
69		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ( bday ‘𝑦) ∈ ( bday ‘𝐴))
70		bdayelon 27715	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅)) ∈ On
71		bdayelon 27715	. . . . . . . . . . . . . . 15 ⊢ ( bday ‘𝐴) ∈ On
72		ontr2 6354	. . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . 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 584	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ( bday ‘((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅)) ∈ ( bday ‘𝐴))
75	45	scutcld 27744	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) ∈ No )
76		oldbday 27846	. . . . . . . . . . . . . 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 587	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) ∈ ( O ‘( bday ‘𝐴)))
79		elons 28190	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ Ons ↔ (𝐴 ∈ No ∧ ( R ‘𝐴) = ∅))
80	79	simprbi 496	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ Ons → ( R ‘𝐴) = ∅)
81	80	ad2antrr 726	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ( R ‘𝐴) = ∅)
82	81	uneq2d 4115	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → (( L ‘𝐴) ∪ ( R ‘𝐴)) = (( L ‘𝐴) ∪ ∅))
83		lrold 27842	. . . . . . . . . . . . 13 ⊢ (( L ‘𝐴) ∪ ( R ‘𝐴)) = ( O ‘( bday ‘𝐴))
84		un0 4341	. . . . . . . . . . . . 13 ⊢ (( L ‘𝐴) ∪ ∅) = ( L ‘𝐴)
85	82, 83, 84	3eqtr3g 2789	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ( O ‘( bday ‘𝐴)) = ( L ‘𝐴))
86	78, 85	eleqtrd 2833	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) ∈ ( L ‘𝐴))
87		leftlt 27808	. . . . . . . . . . 11 ⊢ (((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) ∈ ( L ‘𝐴) → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) <s 𝐴)
88	86, 87	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) <s 𝐴)
89		simplr 768	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → 𝑦 ∈ No )
90		slerflex 27702	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ No → 𝑦 ≤s 𝑦)
91		lrcut 27849	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ No → (( L ‘𝑦) |s ( R ‘𝑦)) = 𝑦)
92	90, 91	breqtrrd 5117	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ No → 𝑦 ≤s (( L ‘𝑦) |s ( R ‘𝑦)))
93	89, 92	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → 𝑦 ≤s (( L ‘𝑦) |s ( R ‘𝑦)))
94		uneq2 4109	. . . . . . . . . . . . . . . 16 ⊢ (( R ‘𝑦) = ∅ → (( L ‘𝑦) ∪ ( R ‘𝑦)) = (( L ‘𝑦) ∪ ∅))
95		un0 4341	. . . . . . . . . . . . . . . 16 ⊢ (( L ‘𝑦) ∪ ∅) = ( L ‘𝑦)
96	94, 95	eqtrdi 2782	. . . . . . . . . . . . . . 15 ⊢ (( R ‘𝑦) = ∅ → (( L ‘𝑦) ∪ ( R ‘𝑦)) = ( L ‘𝑦))
97		eqcom 2738	. . . . . . . . . . . . . . . 16 ⊢ (( R ‘𝑦) = ∅ ↔ ∅ = ( R ‘𝑦))
98	97	biimpi 216	. . . . . . . . . . . . . . 15 ⊢ (( R ‘𝑦) = ∅ → ∅ = ( R ‘𝑦))
99	96, 98	oveq12d 7364	. . . . . . . . . . . . . 14 ⊢ (( R ‘𝑦) = ∅ → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) = (( L ‘𝑦) |s ( R ‘𝑦)))
100	99	breq2d 5101	. . . . . . . . . . . . 13 ⊢ (( R ‘𝑦) = ∅ → (𝑦 ≤s ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) ↔ 𝑦 ≤s (( L ‘𝑦) |s ( R ‘𝑦))))
101	93, 100	imbitrrid 246	. . . . . . . . . . . 12 ⊢ (( R ‘𝑦) = ∅ → (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → 𝑦 ≤s ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅)))
102		simprlr 779	. . . . . . . . . . . . . 14 ⊢ ((( R ‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → 𝑦 ∈ No )
103	75	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((( R ‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) ∈ No )
104		n0 4300	. . . . . . . . . . . . . . . . . 18 ⊢ (( R ‘𝑦) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ ( R ‘𝑦))
105		breq2 5093	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝑤 → (𝑦 ≤s 𝑧 ↔ 𝑦 ≤s 𝑤))
106		elun2 4130	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑤 ∈ ( R ‘𝑦) → 𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
107	106	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ( R ‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → 𝑤 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦)))
108		simprlr 779	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ ( R ‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → 𝑦 ∈ No )
109	37	sseli 3925	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ( R ‘𝑦) → 𝑤 ∈ No )
110	109	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ ( R ‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → 𝑤 ∈ No )
111		rightgt 27809	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑤 ∈ ( R ‘𝑦) → 𝑦 <s 𝑤)
112	111	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑤 ∈ ( R ‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → 𝑦 <s 𝑤)
113	108, 110, 112	sltled 27708	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑤 ∈ ( R ‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → 𝑦 ≤s 𝑤)
114	105, 107, 113	rspcedvdw 3575	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑤 ∈ ( R ‘𝑦) ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → ∃𝑧 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝑦 ≤s 𝑧)
115	114	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ ( R ‘𝑦) → (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ∃𝑧 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝑦 ≤s 𝑧))
116	115	exlimiv 1931	. . . . . . . . . . . . . . . . . 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 217	. . . . . . . . . . . . . . . . 17 ⊢ (( R ‘𝑦) ≠ ∅ → (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ∃𝑧 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝑦 ≤s 𝑧))
118	117	imp 406	. . . . . . . . . . . . . . . 16 ⊢ ((( R ‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → ∃𝑧 ∈ (( L ‘𝑦) ∪ ( R ‘𝑦))𝑦 ≤s 𝑧)
119	118	orcd 873	. . . . . . . . . . . . . . 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		lltropt 27817	. . . . . . . . . . . . . . . . 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 13	. . . . . . . . . . . . . . . 16 ⊢ ((( R ‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → (( L ‘𝑦) ∪ ( R ‘𝑦)) <<s ∅)
123	102, 91	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((( R ‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → (( L ‘𝑦) |s ( R ‘𝑦)) = 𝑦)
124	123	eqcomd 2737	. . . . . . . . . . . . . . . 16 ⊢ ((( R ‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → 𝑦 = (( L ‘𝑦) |s ( R ‘𝑦)))
125		eqidd 2732	. . . . . . . . . . . . . . . 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	sltrecd 27763	. . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . 14 ⊢ ((( R ‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → 𝑦 <s ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅))
128	102, 103, 127	sltled 27708	. . . . . . . . . . . . 13 ⊢ ((( R ‘𝑦) ≠ ∅ ∧ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴))) → 𝑦 ≤s ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅))
129	128	ex 412	. . . . . . . . . . . 12 ⊢ (( R ‘𝑦) ≠ ∅ → (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → 𝑦 ≤s ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅)))
130	101, 129	pm2.61ine 3011	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → 𝑦 ≤s ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅))
131		slenlt 27691	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ¬ ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) <s 𝑦)
134		breq1 5092	. . . . . . . . . . . 12 ⊢ (𝑧 = ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) → (𝑧 <s 𝐴 ↔ ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) <s 𝐴))
135		breq1 5092	. . . . . . . . . . . . 13 ⊢ (𝑧 = ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) → (𝑧 <s 𝑦 ↔ ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) <s 𝑦))
136	135	notbid 318	. . . . . . . . . . . 12 ⊢ (𝑧 = ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) → (¬ 𝑧 <s 𝑦 ↔ ¬ ((( L ‘𝑦) ∪ ( R ‘𝑦)) |s ∅) <s 𝑦))
137	134, 136	anbi12d 632	. . . . . . . . . . 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 3572	. . . . . . . . . 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 836	. . . . . . . . 9 ⊢ (((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) ∧ ( bday ‘𝑦) ∈ ( bday ‘𝐴)) → ∃𝑧 ∈ Ons (𝑧 <s 𝐴 ∧ ¬ 𝑧 <s 𝑦))
140	139	ex 412	. . . . . . . 8 ⊢ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) → (( bday ‘𝑦) ∈ ( bday ‘𝐴) → ∃𝑧 ∈ Ons (𝑧 <s 𝐴 ∧ ¬ 𝑧 <s 𝑦)))
141		ontri1 6340	. . . . . . . . . 10 ⊢ ((( bday ‘𝐴) ∈ On ∧ ( bday ‘𝑦) ∈ On) → (( bday ‘𝐴) ⊆ ( bday ‘𝑦) ↔ ¬ ( bday ‘𝑦) ∈ ( bday ‘𝐴)))
142	71, 51, 141	mp2an 692	. . . . . . . . 9 ⊢ (( bday ‘𝐴) ⊆ ( bday ‘𝑦) ↔ ¬ ( bday ‘𝑦) ∈ ( bday ‘𝐴))
143	142	con2bii 357	. . . . . . . 8 ⊢ (( bday ‘𝑦) ∈ ( bday ‘𝐴) ↔ ¬ ( bday ‘𝐴) ⊆ ( bday ‘𝑦))
144		rexanali 3086	. . . . . . . 8 ⊢ (∃𝑧 ∈ Ons (𝑧 <s 𝐴 ∧ ¬ 𝑧 <s 𝑦) ↔ ¬ ∀𝑧 ∈ Ons (𝑧 <s 𝐴 → 𝑧 <s 𝑦))
145	140, 143, 144	3imtr3g 295	. . . . . . 7 ⊢ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) → (¬ ( bday ‘𝐴) ⊆ ( bday ‘𝑦) → ¬ ∀𝑧 ∈ Ons (𝑧 <s 𝐴 → 𝑧 <s 𝑦)))
146	145	con4d 115	. . . . . 6 ⊢ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) → (∀𝑧 ∈ Ons (𝑧 <s 𝐴 → 𝑧 <s 𝑦) → ( bday ‘𝐴) ⊆ ( bday ‘𝑦)))
147	31, 146	syl5 34	. . . . 5 ⊢ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) → ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} → ( bday ‘𝐴) ⊆ ( bday ‘𝑦)))
148	147	adantrd 491	. . . 4 ⊢ ((𝐴 ∈ Ons ∧ 𝑦 ∈ No ) → (({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} ∧ {𝑦} <<s ∅) → ( bday ‘𝐴) ⊆ ( bday ‘𝑦)))
149	148	ralrimiva 3124	. . 3 ⊢ (𝐴 ∈ Ons → ∀𝑦 ∈ No (({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s {𝑦} ∧ {𝑦} <<s ∅) → ( bday ‘𝐴) ⊆ ( bday ‘𝑦)))
150	3, 8	elpwd 4553	. . . . 5 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∈ 𝒫 No )
151		nulssgt 27739	. . . . 5 ⊢ ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} ∈ 𝒫 No → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s ∅)
152	150, 151	syl 17	. . . 4 ⊢ (𝐴 ∈ Ons → {𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} <<s ∅)
153		eqscut2 27747	. . . 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 584	. . 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 1343	. 2 ⊢ (𝐴 ∈ Ons → ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} |s ∅) = 𝐴)
156	155	eqcomd 2737	1 ⊢ (𝐴 ∈ Ons → 𝐴 = ({𝑥 ∈ Ons ∣ 𝑥 <s 𝐴} |s ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436   ∪ cun 3895   ⊆ wss 3897  ∅c0 4280  𝒫 cpw 4547  {csn 4573   class class class wbr 5089  dom cdm 5614   “ cima 5617  Oncon0 6306  Fun wfun 6475  ‘cfv 6481  (class class class)co 7346   No csur 27578   <s cslt 27579   bday cbday 27580   ≤s csle 27683   <<s csslt 27720   |s cscut 27722   O cold 27784   L cleft 27786   R cright 27787  On_scons 28188

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-2o 8386  df-no 27581  df-slt 27582  df-bday 27583  df-sle 27684  df-sslt 27721  df-scut 27723  df-made 27788  df-old 27789  df-new 27790  df-left 27791  df-right 27792  df-ons 28189

This theorem is referenced by:  bdayon  28209  onsfi  28283  n0cutlt  28285