Theorem oniso 28687

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 28163	. . . . . . 7 ⊢ Fun bday
2		funres 6602	. . . . . . 7 ⊢ (Fun bday → Fun (bday ↾ Ons))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun (bday ↾ Ons)
4		dmres 6015	. . . . . . 7 ⊢ dom (bday ↾ Ons) = (Ons ∩ dom bday)
5		bdaydm 28165	. . . . . . . 8 ⊢ dom bday = No
6	5	ineq2i 4117	. . . . . . 7 ⊢ (Ons ∩ dom bday) = (Ons ∩ No)
7		onssno 28670	. . . . . . . 8 ⊢ Ons ⊆ No
8		dfss2 4097	. . . . . . . 8 ⊢ (Ons ⊆ No ↔ (Ons ∩ No) = Ons)
9	7, 8	mpbi 233	. . . . . . 7 ⊢ (Ons ∩ No) = Ons
10	4, 6, 9	3eqtri 2790	. . . . . 6 ⊢ dom (bday ↾ Ons) = Ons
11		df-fn 6561	. . . . . 6 ⊢ ((bday ↾ Ons) Fn Ons ↔ (Fun (bday ↾ Ons) ∧ dom (bday ↾ Ons) = Ons))
12	3, 10, 11	mpbir2an 724	. . . . 5 ⊢ (bday ↾ Ons) Fn Ons
13		rnresss 6018	. . . . . 6 ⊢ ran (bday ↾ Ons) ⊆ ran bday
14		bdayrn 28167	. . . . . 6 ⊢ ran bday = On
15	13, 14	sseqtri 3961	. . . . 5 ⊢ ran (bday ↾ Ons) ⊆ On
16		df-f 6562	. . . . 5 ⊢ ((bday ↾ Ons):Ons⟶On ↔ ((bday ↾ Ons) Fn Ons ∧ ran (bday ↾ Ons) ⊆ On))
17	12, 15, 16	mpbir2an 724	. . . 4 ⊢ (bday ↾ Ons):Ons⟶On
18		fvres 6927	. . . . . . 7 ⊢ (𝑥 ∈ Ons → ((bday ↾ Ons)‘𝑥) = (bday‘𝑥))
19		fvres 6927	. . . . . . 7 ⊢ (𝑦 ∈ Ons → ((bday ↾ Ons)‘𝑦) = (bday‘𝑦))
20	18, 19	eqeqan12d 2777	. . . . . 6 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (((bday ↾ Ons)‘𝑥) = ((bday ↾ Ons)‘𝑦) ↔ (bday‘𝑥) = (bday‘𝑦)))
21		bday11on 28681	. . . . . . 7 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons ∧ (bday‘𝑥) = (bday‘𝑦)) → 𝑥 = 𝑦)
22	21	3expia 1139	. . . . . 6 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → ((bday‘𝑥) = (bday‘𝑦) → 𝑥 = 𝑦))
23	20, 22	sylbid 243	. . . . 5 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (((bday ↾ Ons)‘𝑥) = ((bday ↾ Ons)‘𝑦) → 𝑥 = 𝑦))
24	23	rgen2 3206	. . . 4 ⊢ ∀𝑥 ∈ Ons ∀𝑦 ∈ Ons (((bday ↾ Ons)‘𝑥) = ((bday ↾ Ons)‘𝑦) → 𝑥 = 𝑦)
25		dff13 7281	. . . 4 ⊢ ((bday ↾ Ons):Ons⟶1-1On ↔ ((bday ↾ Ons):Ons⟶On ∧ ∀𝑥 ∈ Ons ∀𝑦 ∈ Ons (((bday ↾ Ons)‘𝑥) = ((bday ↾ Ons)‘𝑦) → 𝑥 = 𝑦)))
26	17, 24, 25	mpbir2an 724	. . 3 ⊢ (bday ↾ Ons):Ons⟶1-1On
27		fveqeq2 6917	. . . . . . . 8 ⊢ (𝑦 = ((O‘𝑥) |s ∅) → (((bday ↾ Ons)‘𝑦) = 𝑥 ↔ ((bday ↾ Ons)‘((O‘𝑥) |s ∅)) = 𝑥))
28		fvex 6921	. . . . . . . . . 10 ⊢ (O‘𝑥) ∈ V
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (O‘𝑥) ∈ V)
30		oldssno 28257	. . . . . . . . . 10 ⊢ (O‘𝑥) ⊆ No
31	30	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (O‘𝑥) ⊆ No)
32		eqidd 2764	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ((O‘𝑥) |s ∅) = ((O‘𝑥) |s ∅))
33	29, 31, 32	elons2d 28675	. . . . . . . 8 ⊢ (𝑥 ∈ On → ((O‘𝑥) |s ∅) ∈ Ons)
34	33	fvresd 6928	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ((bday ↾ Ons)‘((O‘𝑥) |s ∅)) = (bday‘((O‘𝑥) |s ∅)))
35	28	elpw 4564	. . . . . . . . . . . . 13 ⊢ ((O‘𝑥) ∈ 𝒫 No ↔ (O‘𝑥) ⊆ No)
36	30, 35	mpbir 234	. . . . . . . . . . . 12 ⊢ (O‘𝑥) ∈ 𝒫 No
37		nulsgts 28192	. . . . . . . . . . . 12 ⊢ ((O‘𝑥) ∈ 𝒫 No → (O‘𝑥) <<s ∅)
38	36, 37	ax-mp 5	. . . . . . . . . . 11 ⊢ (O‘𝑥) <<s ∅
39		id 23	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ∈ On)
40		un0 4347	. . . . . . . . . . . . 13 ⊢ ((O‘𝑥) ∪ ∅) = (O‘𝑥)
41	40	imaeq2i 6062	. . . . . . . . . . . 12 ⊢ (bday “ ((O‘𝑥) ∪ ∅)) = (bday “ (O‘𝑥))
42		oldbdayim 28305	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (O‘𝑥) → (bday‘𝑦) ∈ 𝑥)
43	42	rgen 3082	. . . . . . . . . . . . . 14 ⊢ ∀𝑦 ∈ (O‘𝑥)(bday‘𝑦) ∈ 𝑥
44	43	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ (O‘𝑥)(bday‘𝑦) ∈ 𝑥)
45	30, 5	sseqtrri 3962	. . . . . . . . . . . . . 14 ⊢ (O‘𝑥) ⊆ dom bday
46		funimass4 6972	. . . . . . . . . . . . . 14 ⊢ ((Fun bday ∧ (O‘𝑥) ⊆ dom bday) → ((bday “ (O‘𝑥)) ⊆ 𝑥 ↔ ∀𝑦 ∈ (O‘𝑥)(bday‘𝑦) ∈ 𝑥))
47	1, 45, 46	mp2an 705	. . . . . . . . . . . . 13 ⊢ ((bday “ (O‘𝑥)) ⊆ 𝑥 ↔ ∀𝑦 ∈ (O‘𝑥)(bday‘𝑦) ∈ 𝑥)
48	44, 47	sylibr 237	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (bday “ (O‘𝑥)) ⊆ 𝑥)
49	41, 48	eqsstrid 3951	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → (bday “ ((O‘𝑥) ∪ ∅)) ⊆ 𝑥)
50		cutbdaybnd 28211	. . . . . . . . . . 11 ⊢ (((O‘𝑥) <<s ∅ ∧ 𝑥 ∈ On ∧ (bday “ ((O‘𝑥) ∪ ∅)) ⊆ 𝑥) → (bday‘((O‘𝑥) |s ∅)) ⊆ 𝑥)
51	38, 39, 49, 50	mp3an2i 1496	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → (bday‘((O‘𝑥) |s ∅)) ⊆ 𝑥)
52		sltssep 28183	. . . . . . . . . . . . . . . 16 ⊢ ((O‘𝑥) <<s {𝑤} → ∀𝑦 ∈ (O‘𝑥)∀𝑧 ∈ {𝑤}𝑦 <s 𝑧)
53		vex 3458	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑤 ∈ V
54		breq2 5110	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝑤))
55	53, 54	ralsn 4642	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧 ∈ {𝑤}𝑦 <s 𝑧 ↔ 𝑦 <s 𝑤)
56	55	ralbii 3112	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ (O‘𝑥)∀𝑧 ∈ {𝑤}𝑦 <s 𝑧 ↔ ∀𝑦 ∈ (O‘𝑥)𝑦 <s 𝑤)
57	52, 56	sylib 221	. . . . . . . . . . . . . . 15 ⊢ ((O‘𝑥) <<s {𝑤} → ∀𝑦 ∈ (O‘𝑥)𝑦 <s 𝑤)
58		ltsirr 28133	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ No → ¬ 𝑤 <s 𝑤)
59	58	3ad2ant2 1152	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ (O‘𝑥)𝑦 <s 𝑤) → ¬ 𝑤 <s 𝑤)
60		oldbday 28317	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No) → (𝑤 ∈ (O‘𝑥) ↔ (bday‘𝑤) ∈ 𝑥))
61	60	3adant3 1150	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ (O‘𝑥)𝑦 <s 𝑤) → (𝑤 ∈ (O‘𝑥) ↔ (bday‘𝑤) ∈ 𝑥))
62		breq1 5109	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑤 → (𝑦 <s 𝑤 ↔ 𝑤 <s 𝑤))
63	62	rspccv 3577	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑦 ∈ (O‘𝑥)𝑦 <s 𝑤 → (𝑤 ∈ (O‘𝑥) → 𝑤 <s 𝑤))
64	63	3ad2ant3 1153	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ (O‘𝑥)𝑦 <s 𝑤) → (𝑤 ∈ (O‘𝑥) → 𝑤 <s 𝑤))
65	61, 64	sylbird 263	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ (O‘𝑥)𝑦 <s 𝑤) → ((bday‘𝑤) ∈ 𝑥 → 𝑤 <s 𝑤))
66	59, 65	mtod 201	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ (O‘𝑥)𝑦 <s 𝑤) → ¬ (bday‘𝑤) ∈ 𝑥)
67		simp1 1154	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ (O‘𝑥)𝑦 <s 𝑤) → 𝑥 ∈ On)
68		bdayon 28168	. . . . . . . . . . . . . . . . . 18 ⊢ (bday‘𝑤) ∈ On
69		ontri1 6415	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ (bday‘𝑤) ∈ On) → (𝑥 ⊆ (bday‘𝑤) ↔ ¬ (bday‘𝑤) ∈ 𝑥))
70	67, 68, 69	sylancl 598	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ (O‘𝑥)𝑦 <s 𝑤) → (𝑥 ⊆ (bday‘𝑤) ↔ ¬ (bday‘𝑤) ∈ 𝑥))
71	66, 70	mpbird 260	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ (O‘𝑥)𝑦 <s 𝑤) → 𝑥 ⊆ (bday‘𝑤))
72	71	3expia 1139	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No) → (∀𝑦 ∈ (O‘𝑥)𝑦 <s 𝑤 → 𝑥 ⊆ (bday‘𝑤)))
73	57, 72	syl5 35	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No) → ((O‘𝑥) <<s {𝑤} → 𝑥 ⊆ (bday‘𝑤)))
74	73	adantrd 497	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No) → (((O‘𝑥) <<s {𝑤} ∧ {𝑤} <<s ∅) → 𝑥 ⊆ (bday‘𝑤)))
75	74	ralrimiva 3158	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → ∀𝑤 ∈ No (((O‘𝑥) <<s {𝑤} ∧ {𝑤} <<s ∅) → 𝑥 ⊆ (bday‘𝑤)))
76		ssint 4927	. . . . . . . . . . . . 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 28164	. . . . . . . . . . . . . 14 ⊢ bday Fn No
78		ssrab2 4008	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ No ∣ ((O‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)} ⊆ No
79		sseq2 3939	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (bday‘𝑤) → (𝑥 ⊆ 𝑧 ↔ 𝑥 ⊆ (bday‘𝑤)))
80	79	ralima 7266	. . . . . . . . . . . . . 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 705	. . . . . . . . . . . . 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 4594	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
83	82	breq2d 5118	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ((O‘𝑥) <<s {𝑦} ↔ (O‘𝑥) <<s {𝑤}))
84	82	breq1d 5116	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ({𝑦} <<s ∅ ↔ {𝑤} <<s ∅))
85	83, 84	anbi12d 644	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → (((O‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅) ↔ ((O‘𝑥) <<s {𝑤} ∧ {𝑤} <<s ∅)))
86	85	ralrab 3655	. . . . . . . . . . . . 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 300	. . . . . . . . . . . 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 237	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ⊆ ∩ (bday “ {𝑦 ∈ No ∣ ((O‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)}))
89		cutbday 28200	. . . . . . . . . . . 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 3954	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ (bday‘((O‘𝑥) |s ∅)))
92	51, 91	eqssd 3930	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (bday‘((O‘𝑥) |s ∅)) = 𝑥)
93	34, 92	eqtrd 2798	. . . . . . . 8 ⊢ (𝑥 ∈ On → ((bday ↾ Ons)‘((O‘𝑥) |s ∅)) = 𝑥)
94	27, 33, 93	rspcedvdw 3583	. . . . . . 7 ⊢ (𝑥 ∈ On → ∃𝑦 ∈ Ons ((bday ↾ Ons)‘𝑦) = 𝑥)
95		fvelrnb 6968	. . . . . . . 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 237	. . . . . 6 ⊢ (𝑥 ∈ On → 𝑥 ∈ ran (bday ↾ Ons))
98	97	ssriv 3917	. . . . 5 ⊢ On ⊆ ran (bday ↾ Ons)
99	15, 98	eqssi 3929	. . . 4 ⊢ ran (bday ↾ Ons) = On
100		df-fo 6564	. . . 4 ⊢ ((bday ↾ Ons):Ons⟶ontoOn ↔ ((bday ↾ Ons) Fn Ons ∧ ran (bday ↾ Ons) = On))
101	12, 99, 100	mpbir2an 724	. . 3 ⊢ (bday ↾ Ons):Ons⟶ontoOn
102		df-f1o 6565	. . 3 ⊢ ((bday ↾ Ons):Ons⟶1-1ontoOn ↔ ((bday ↾ Ons):Ons⟶1-1On ∧ (bday ↾ Ons):Ons⟶ontoOn))
103	26, 101, 102	mpbir2an 724	. 2 ⊢ (bday ↾ Ons):Ons⟶1-1ontoOn
104		onlts 28683	. . . . 5 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (𝑥 <s 𝑦 ↔ (bday‘𝑥) ∈ (bday‘𝑦)))
105		fvex 6921	. . . . . 6 ⊢ (bday‘𝑦) ∈ V
106	105	epeli 5557	. . . . 5 ⊢ ((bday‘𝑥) E (bday‘𝑦) ↔ (bday‘𝑥) ∈ (bday‘𝑦))
107	104, 106	bitr4di 292	. . . 4 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (𝑥 <s 𝑦 ↔ (bday‘𝑥) E (bday‘𝑦)))
108	18, 19	breqan12d 5122	. . . 4 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (((bday ↾ Ons)‘𝑥) E ((bday ↾ Ons)‘𝑦) ↔ (bday‘𝑥) E (bday‘𝑦)))
109	107, 108	bitr4d 285	. . 3 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (𝑥 <s 𝑦 ↔ ((bday ↾ Ons)‘𝑥) E ((bday ↾ Ons)‘𝑦)))
110	109	rgen2 3206	. 2 ⊢ ∀𝑥 ∈ Ons ∀𝑦 ∈ Ons (𝑥 <s 𝑦 ↔ ((bday ↾ Ons)‘𝑥) E ((bday ↾ Ons)‘𝑦))
111		df-isom 6567	. 2 ⊢ ((bday ↾ Ons) Isom <s , E (Ons, On) ↔ ((bday ↾ Ons):Ons⟶1-1ontoOn ∧ ∀𝑥 ∈ Ons ∀𝑦 ∈ Ons (𝑥 <s 𝑦 ↔ ((bday ↾ Ons)‘𝑥) E ((bday ↾ Ons)‘𝑦))))
112	103, 110, 111	mpbir2an 724	1 ⊢ (bday ↾ Ons) Isom <s , E (Ons, On)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  𝒫 cpw 4560  {csn 4588  ∩ cint 4910   class class class wbr 5106   E cep 5554  dom cdm 5655  ran crn 5656   ↾ cres 5657   “ cima 5658  Oncon0 6380  Fun wfun 6552   Fn wfn 6553  ⟶wf 6554  ⟶1-1wf1 6555  ⟶ontowfo 6556  ⟶1-1ontowf1o 6557  ‘cfv 6558   Isom wiso 6559  (class class class)co 7443  Nocsur 28027   <s clts 28028  bdaycbday 28029   <<s cslts 28173   |s ccuts 28175  Ocold 28239  On_scons 28667

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-isom 6567  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-ons 28668

This theorem is used by:  onswe  28688  onsse  28689  addonbday  28695