Theorem oniso 28341

Description: The birthday function restricted to the surreal ordinals forms an order-preserving isomorphism with the regular ordinals. (Contributed by Scott Fenton, 8-Nov-2025.)

Assertion

Ref	Expression
oniso	⊢ ( bday ↾ Ons) Isom <s , E (Ons, On)

Proof of Theorem oniso

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

Step	Hyp	Ref	Expression
1		bdayfun 27817	. . . . . . 7 ⊢ Fun bday
2		funres 6559	. . . . . . 7 ⊢ (Fun bday → Fun ( bday ↾ Ons))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun ( bday ↾ Ons)
4		dmres 5996	. . . . . . 7 ⊢ dom ( bday ↾ Ons) = (Ons ∩ dom bday )
5		bdaydm 27819	. . . . . . . 8 ⊢ dom bday = No
6	5	ineq2i 4169	. . . . . . 7 ⊢ (Ons ∩ dom bday ) = (Ons ∩ No )
7		onssno 28324	. . . . . . . 8 ⊢ Ons ⊆ No
8		dfss2 3922	. . . . . . . 8 ⊢ (Ons ⊆ No ↔ (Ons ∩ No ) = Ons)
9	7, 8	mpbi 232	. . . . . . 7 ⊢ (Ons ∩ No ) = Ons
10	4, 6, 9	3eqtri 2788	. . . . . 6 ⊢ dom ( bday ↾ Ons) = Ons
11		df-fn 6520	. . . . . 6 ⊢ (( bday ↾ Ons) Fn Ons ↔ (Fun ( bday ↾ Ons) ∧ dom ( bday ↾ Ons) = Ons))
12	3, 10, 11	mpbir2an 721	. . . . 5 ⊢ ( bday ↾ Ons) Fn Ons
13		rnresss 6001	. . . . . 6 ⊢ ran ( bday ↾ Ons) ⊆ ran bday
14		bdayrn 27821	. . . . . 6 ⊢ ran bday = On
15	13, 14	sseqtri 3984	. . . . 5 ⊢ ran ( bday ↾ Ons) ⊆ On
16		df-f 6521	. . . . 5 ⊢ (( bday ↾ Ons):Ons⟶On ↔ (( bday ↾ Ons) Fn Ons ∧ ran ( bday ↾ Ons) ⊆ On))
17	12, 15, 16	mpbir2an 721	. . . 4 ⊢ ( bday ↾ Ons):Ons⟶On
18		fvres 6882	. . . . . . 7 ⊢ (𝑥 ∈ Ons → (( bday ↾ Ons)‘𝑥) = ( bday ‘𝑥))
19		fvres 6882	. . . . . . 7 ⊢ (𝑦 ∈ Ons → (( bday ↾ Ons)‘𝑦) = ( bday ‘𝑦))
20	18, 19	eqeqan12d 2775	. . . . . 6 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → ((( bday ↾ Ons)‘𝑥) = (( bday ↾ Ons)‘𝑦) ↔ ( bday ‘𝑥) = ( bday ‘𝑦)))
21		bday11on 28335	. . . . . . 7 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons ∧ ( bday ‘𝑥) = ( bday ‘𝑦)) → 𝑥 = 𝑦)
22	21	3expia 1133	. . . . . 6 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (( bday ‘𝑥) = ( bday ‘𝑦) → 𝑥 = 𝑦))
23	20, 22	sylbid 242	. . . . 5 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → ((( bday ↾ Ons)‘𝑥) = (( bday ↾ Ons)‘𝑦) → 𝑥 = 𝑦))
24	23	rgen2 3201	. . . 4 ⊢ ∀𝑥 ∈ Ons ∀𝑦 ∈ Ons ((( bday ↾ Ons)‘𝑥) = (( bday ↾ Ons)‘𝑦) → 𝑥 = 𝑦)
25		dff13 7234	. . . 4 ⊢ (( bday ↾ Ons):Ons–1-1→On ↔ (( bday ↾ Ons):Ons⟶On ∧ ∀𝑥 ∈ Ons ∀𝑦 ∈ Ons ((( bday ↾ Ons)‘𝑥) = (( bday ↾ Ons)‘𝑦) → 𝑥 = 𝑦)))
26	17, 24, 25	mpbir2an 721	. . 3 ⊢ ( bday ↾ Ons):Ons–1-1→On
27		fveqeq2 6872	. . . . . . . 8 ⊢ (𝑦 = (( O ‘𝑥) |s ∅) → ((( bday ↾ Ons)‘𝑦) = 𝑥 ↔ (( bday ↾ Ons)‘(( O ‘𝑥) |s ∅)) = 𝑥))
28		fvex 6876	. . . . . . . . . 10 ⊢ ( O ‘𝑥) ∈ V
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ( O ‘𝑥) ∈ V)
30		oldssno 27911	. . . . . . . . . 10 ⊢ ( O ‘𝑥) ⊆ No
31	30	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ( O ‘𝑥) ⊆ No )
32		eqidd 2762	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (( O ‘𝑥) |s ∅) = (( O ‘𝑥) |s ∅))
33	29, 31, 32	elons2d 28329	. . . . . . . 8 ⊢ (𝑥 ∈ On → (( O ‘𝑥) |s ∅) ∈ Ons)
34	33	fvresd 6883	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (( bday ↾ Ons)‘(( O ‘𝑥) |s ∅)) = ( bday ‘(( O ‘𝑥) |s ∅)))
35	28	elpw 4558	. . . . . . . . . . . . 13 ⊢ (( O ‘𝑥) ∈ 𝒫 No ↔ ( O ‘𝑥) ⊆ No )
36	30, 35	mpbir 233	. . . . . . . . . . . 12 ⊢ ( O ‘𝑥) ∈ 𝒫 No
37		nulsgts 27846	. . . . . . . . . . . 12 ⊢ (( O ‘𝑥) ∈ 𝒫 No → ( O ‘𝑥) <<s ∅)
38	36, 37	ax-mp 5	. . . . . . . . . . 11 ⊢ ( O ‘𝑥) <<s ∅
39		id 22	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ∈ On)
40		un0 4347	. . . . . . . . . . . . 13 ⊢ (( O ‘𝑥) ∪ ∅) = ( O ‘𝑥)
41	40	imaeq2i 6044	. . . . . . . . . . . 12 ⊢ ( bday “ (( O ‘𝑥) ∪ ∅)) = ( bday “ ( O ‘𝑥))
42		oldbdayim 27959	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ( O ‘𝑥) → ( bday ‘𝑦) ∈ 𝑥)
43	42	rgen 3077	. . . . . . . . . . . . . 14 ⊢ ∀𝑦 ∈ ( O ‘𝑥)( bday ‘𝑦) ∈ 𝑥
44	43	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ ( O ‘𝑥)( bday ‘𝑦) ∈ 𝑥)
45	30, 5	sseqtrri 3985	. . . . . . . . . . . . . 14 ⊢ ( O ‘𝑥) ⊆ dom bday
46		funimass4 6927	. . . . . . . . . . . . . 14 ⊢ ((Fun bday ∧ ( O ‘𝑥) ⊆ dom bday ) → (( bday “ ( O ‘𝑥)) ⊆ 𝑥 ↔ ∀𝑦 ∈ ( O ‘𝑥)( bday ‘𝑦) ∈ 𝑥))
47	1, 45, 46	mp2an 702	. . . . . . . . . . . . 13 ⊢ (( bday “ ( O ‘𝑥)) ⊆ 𝑥 ↔ ∀𝑦 ∈ ( O ‘𝑥)( bday ‘𝑦) ∈ 𝑥)
48	44, 47	sylibr 236	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → ( bday “ ( O ‘𝑥)) ⊆ 𝑥)
49	41, 48	eqsstrid 3974	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ( bday “ (( O ‘𝑥) ∪ ∅)) ⊆ 𝑥)
50		cutbdaybnd 27865	. . . . . . . . . . 11 ⊢ ((( O ‘𝑥) <<s ∅ ∧ 𝑥 ∈ On ∧ ( bday “ (( O ‘𝑥) ∪ ∅)) ⊆ 𝑥) → ( bday ‘(( O ‘𝑥) |s ∅)) ⊆ 𝑥)
51	38, 39, 49, 50	mp3an2i 1486	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ( bday ‘(( O ‘𝑥) |s ∅)) ⊆ 𝑥)
52		sltssep 27837	. . . . . . . . . . . . . . . 16 ⊢ (( O ‘𝑥) <<s {𝑤} → ∀𝑦 ∈ ( O ‘𝑥)∀𝑧 ∈ {𝑤}𝑦 <s 𝑧)
53		vex 3457	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑤 ∈ V
54		breq2 5103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝑤))
55	53, 54	ralsn 4639	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧 ∈ {𝑤}𝑦 <s 𝑧 ↔ 𝑦 <s 𝑤)
56	55	ralbii 3107	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ ( O ‘𝑥)∀𝑧 ∈ {𝑤}𝑦 <s 𝑧 ↔ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤)
57	52, 56	sylib 220	. . . . . . . . . . . . . . 15 ⊢ (( O ‘𝑥) <<s {𝑤} → ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤)
58		ltsirr 27787	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ No → ¬ 𝑤 <s 𝑤)
59	58	3ad2ant2 1146	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → ¬ 𝑤 <s 𝑤)
60		oldbday 27971	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ) → (𝑤 ∈ ( O ‘𝑥) ↔ ( bday ‘𝑤) ∈ 𝑥))
61	60	3adant3 1144	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → (𝑤 ∈ ( O ‘𝑥) ↔ ( bday ‘𝑤) ∈ 𝑥))
62		breq1 5102	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑤 → (𝑦 <s 𝑤 ↔ 𝑤 <s 𝑤))
63	62	rspccv 3578	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤 → (𝑤 ∈ ( O ‘𝑥) → 𝑤 <s 𝑤))
64	63	3ad2ant3 1147	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → (𝑤 ∈ ( O ‘𝑥) → 𝑤 <s 𝑤))
65	61, 64	sylbird 262	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → (( bday ‘𝑤) ∈ 𝑥 → 𝑤 <s 𝑤))
66	59, 65	mtod 200	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → ¬ ( bday ‘𝑤) ∈ 𝑥)
67		simp1 1148	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → 𝑥 ∈ On)
68		bdayon 27822	. . . . . . . . . . . . . . . . . 18 ⊢ ( bday ‘𝑤) ∈ On
69		ontri1 6376	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ ( bday ‘𝑤) ∈ On) → (𝑥 ⊆ ( bday ‘𝑤) ↔ ¬ ( bday ‘𝑤) ∈ 𝑥))
70	67, 68, 69	sylancl 595	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → (𝑥 ⊆ ( bday ‘𝑤) ↔ ¬ ( bday ‘𝑤) ∈ 𝑥))
71	66, 70	mpbird 259	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → 𝑥 ⊆ ( bday ‘𝑤))
72	71	3expia 1133	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ) → (∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤 → 𝑥 ⊆ ( bday ‘𝑤)))
73	57, 72	syl5 34	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ) → (( O ‘𝑥) <<s {𝑤} → 𝑥 ⊆ ( bday ‘𝑤)))
74	73	adantrd 495	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ) → ((( O ‘𝑥) <<s {𝑤} ∧ {𝑤} <<s ∅) → 𝑥 ⊆ ( bday ‘𝑤)))
75	74	ralrimiva 3153	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → ∀𝑤 ∈ No ((( O ‘𝑥) <<s {𝑤} ∧ {𝑤} <<s ∅) → 𝑥 ⊆ ( bday ‘𝑤)))
76		ssint 4921	. . . . . . . . . . . . 13 ⊢ (𝑥 ⊆ ∩ ( bday “ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)}) ↔ ∀𝑧 ∈ ( bday “ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)})𝑥 ⊆ 𝑧)
77		bdayfn 27818	. . . . . . . . . . . . . 14 ⊢ bday Fn No
78		ssrab2 4033	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)} ⊆ No
79		sseq2 3962	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ( bday ‘𝑤) → (𝑥 ⊆ 𝑧 ↔ 𝑥 ⊆ ( bday ‘𝑤)))
80	79	ralima 7217	. . . . . . . . . . . . . 14 ⊢ (( bday Fn No ∧ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)} ⊆ No ) → (∀𝑧 ∈ ( bday “ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)})𝑥 ⊆ 𝑧 ↔ ∀𝑤 ∈ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)}𝑥 ⊆ ( bday ‘𝑤)))
81	77, 78, 80	mp2an 702	. . . . . . . . . . . . 13 ⊢ (∀𝑧 ∈ ( bday “ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)})𝑥 ⊆ 𝑧 ↔ ∀𝑤 ∈ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)}𝑥 ⊆ ( bday ‘𝑤))
82		sneq 4591	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
83	82	breq2d 5111	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → (( O ‘𝑥) <<s {𝑦} ↔ ( O ‘𝑥) <<s {𝑤}))
84	82	breq1d 5109	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ({𝑦} <<s ∅ ↔ {𝑤} <<s ∅))
85	83, 84	anbi12d 641	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → ((( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅) ↔ (( O ‘𝑥) <<s {𝑤} ∧ {𝑤} <<s ∅)))
86	85	ralrab 3656	. . . . . . . . . . . . 13 ⊢ (∀𝑤 ∈ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)}𝑥 ⊆ ( bday ‘𝑤) ↔ ∀𝑤 ∈ No ((( O ‘𝑥) <<s {𝑤} ∧ {𝑤} <<s ∅) → 𝑥 ⊆ ( bday ‘𝑤)))
87	76, 81, 86	3bitri 299	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ ∩ ( bday “ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)}) ↔ ∀𝑤 ∈ No ((( O ‘𝑥) <<s {𝑤} ∧ {𝑤} <<s ∅) → 𝑥 ⊆ ( bday ‘𝑤)))
88	75, 87	sylibr 236	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ⊆ ∩ ( bday “ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)}))
89		cutbday 27854	. . . . . . . . . . . 12 ⊢ (( O ‘𝑥) <<s ∅ → ( bday ‘(( O ‘𝑥) |s ∅)) = ∩ ( bday “ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)}))
90	38, 89	ax-mp 5	. . . . . . . . . . 11 ⊢ ( bday ‘(( O ‘𝑥) |s ∅)) = ∩ ( bday “ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)})
91	88, 90	sseqtrrdi 3977	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ ( bday ‘(( O ‘𝑥) |s ∅)))
92	51, 91	eqssd 3953	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ( bday ‘(( O ‘𝑥) |s ∅)) = 𝑥)
93	34, 92	eqtrd 2796	. . . . . . . 8 ⊢ (𝑥 ∈ On → (( bday ↾ Ons)‘(( O ‘𝑥) |s ∅)) = 𝑥)
94	27, 33, 93	rspcedvdw 3584	. . . . . . 7 ⊢ (𝑥 ∈ On → ∃𝑦 ∈ Ons (( bday ↾ Ons)‘𝑦) = 𝑥)
95		fvelrnb 6923	. . . . . . . 8 ⊢ (( bday ↾ Ons) Fn Ons → (𝑥 ∈ ran ( bday ↾ Ons) ↔ ∃𝑦 ∈ Ons (( bday ↾ Ons)‘𝑦) = 𝑥))
96	12, 95	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ ran ( bday ↾ Ons) ↔ ∃𝑦 ∈ Ons (( bday ↾ Ons)‘𝑦) = 𝑥)
97	94, 96	sylibr 236	. . . . . 6 ⊢ (𝑥 ∈ On → 𝑥 ∈ ran ( bday ↾ Ons))
98	97	ssriv 3940	. . . . 5 ⊢ On ⊆ ran ( bday ↾ Ons)
99	15, 98	eqssi 3952	. . . 4 ⊢ ran ( bday ↾ Ons) = On
100		df-fo 6523	. . . 4 ⊢ (( bday ↾ Ons):Ons–onto→On ↔ (( bday ↾ Ons) Fn Ons ∧ ran ( bday ↾ Ons) = On))
101	12, 99, 100	mpbir2an 721	. . 3 ⊢ ( bday ↾ Ons):Ons–onto→On
102		df-f1o 6524	. . 3 ⊢ (( bday ↾ Ons):Ons–1-1-onto→On ↔ (( bday ↾ Ons):Ons–1-1→On ∧ ( bday ↾ Ons):Ons–onto→On))
103	26, 101, 102	mpbir2an 721	. 2 ⊢ ( bday ↾ Ons):Ons–1-1-onto→On
104		onlts 28337	. . . . 5 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (𝑥 <s 𝑦 ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑦)))
105		fvex 6876	. . . . . 6 ⊢ ( bday ‘𝑦) ∈ V
106	105	epeli 5547	. . . . 5 ⊢ (( bday ‘𝑥) E ( bday ‘𝑦) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑦))
107	104, 106	bitr4di 291	. . . 4 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (𝑥 <s 𝑦 ↔ ( bday ‘𝑥) E ( bday ‘𝑦)))
108	18, 19	breqan12d 5115	. . . 4 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → ((( bday ↾ Ons)‘𝑥) E (( bday ↾ Ons)‘𝑦) ↔ ( bday ‘𝑥) E ( bday ‘𝑦)))
109	107, 108	bitr4d 284	. . 3 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (𝑥 <s 𝑦 ↔ (( bday ↾ Ons)‘𝑥) E (( bday ↾ Ons)‘𝑦)))
110	109	rgen2 3201	. 2 ⊢ ∀𝑥 ∈ Ons ∀𝑦 ∈ Ons (𝑥 <s 𝑦 ↔ (( bday ↾ Ons)‘𝑥) E (( bday ↾ Ons)‘𝑦))
111		df-isom 6526	. 2 ⊢ (( bday ↾ Ons) Isom <s , E (Ons, On) ↔ (( bday ↾ Ons):Ons–1-1-onto→On ∧ ∀𝑥 ∈ Ons ∀𝑦 ∈ Ons (𝑥 <s 𝑦 ↔ (( bday ↾ Ons)‘𝑥) E (( bday ↾ Ons)‘𝑦))))
112	103, 110, 111	mpbir2an 721	1 ⊢ ( bday ↾ Ons) Isom <s , E (Ons, On)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  {crab 3413  Vcvv 3453   ∪ cun 3902   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  𝒫 cpw 4554  {csn 4581  ∩ cint 4904   class class class wbr 5099   E cep 5544  dom cdm 5645  ran crn 5646   ↾ cres 5647   “ cima 5648  Oncon0 6342  Fun wfun 6511   Fn wfn 6512  ⟶wf 6513  –1-1→wf1 6514  –onto→wfo 6515  –1-1-onto→wf1o 6516  ‘cfv 6517   Isom wiso 6518  (class class class)co 7392   No csur 27681   <s clts 27682   bday cbday 27683   <<s cslts 27827   |s ccuts 27829   O cold 27893  On_scons 28321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-iun 4950  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-pred 6284  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-isom 6526  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-2nd 7967  df-frecs 8257  df-wrecs 8288  df-recs 8337  df-1o 8432  df-2o 8433  df-no 27684  df-lts 27685  df-bday 27686  df-les 27786  df-slts 27828  df-cuts 27830  df-made 27897  df-old 27898  df-left 27900  df-right 27901  df-ons 28322

This theorem is referenced by:  onswe  28342  onsse  28343  addonbday  28349