Theorem oniso 28501

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 27977	. . . . . . 7 ⊢ Fun bday
2		funres 6585	. . . . . . 7 ⊢ (Fun bday → Fun ( bday ↾ Ons))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun ( bday ↾ Ons)
4		dmres 6016	. . . . . . 7 ⊢ dom ( bday ↾ Ons) = (Ons ∩ dom bday )
5		bdaydm 27979	. . . . . . . 8 ⊢ dom bday = No
6	5	ineq2i 4173	. . . . . . 7 ⊢ (Ons ∩ dom bday ) = (Ons ∩ No )
7		onssno 28484	. . . . . . . 8 ⊢ Ons ⊆ No
8		dfss2 3926	. . . . . . . 8 ⊢ (Ons ⊆ No ↔ (Ons ∩ No ) = Ons)
9	7, 8	mpbi 233	. . . . . . 7 ⊢ (Ons ∩ No ) = Ons
10	4, 6, 9	3eqtri 2793	. . . . . 6 ⊢ dom ( bday ↾ Ons) = Ons
11		df-fn 6546	. . . . . 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 6021	. . . . . 6 ⊢ ran ( bday ↾ Ons) ⊆ ran bday
14		bdayrn 27981	. . . . . 6 ⊢ ran bday = On
15	13, 14	sseqtri 3988	. . . . 5 ⊢ ran ( bday ↾ Ons) ⊆ On
16		df-f 6547	. . . . 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 6907	. . . . . . 7 ⊢ (𝑥 ∈ Ons → (( bday ↾ Ons)‘𝑥) = ( bday ‘𝑥))
19		fvres 6907	. . . . . . 7 ⊢ (𝑦 ∈ Ons → (( bday ↾ Ons)‘𝑦) = ( bday ‘𝑦))
20	18, 19	eqeqan12d 2780	. . . . . 6 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → ((( bday ↾ Ons)‘𝑥) = (( bday ↾ Ons)‘𝑦) ↔ ( bday ‘𝑥) = ( bday ‘𝑦)))
21		bday11on 28495	. . . . . . 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 3208	. . . 4 ⊢ ∀𝑥 ∈ Ons ∀𝑦 ∈ Ons ((( bday ↾ Ons)‘𝑥) = (( bday ↾ Ons)‘𝑦) → 𝑥 = 𝑦)
25		dff13 7259	. . . 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 724	. . 3 ⊢ ( bday ↾ Ons):Ons–1-1→On
27		fveqeq2 6897	. . . . . . . 8 ⊢ (𝑦 = (( O ‘𝑥) |s ∅) → ((( bday ↾ Ons)‘𝑦) = 𝑥 ↔ (( bday ↾ Ons)‘(( O ‘𝑥) |s ∅)) = 𝑥))
28		fvex 6901	. . . . . . . . . 10 ⊢ ( O ‘𝑥) ∈ V
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ( O ‘𝑥) ∈ V)
30		oldssno 28071	. . . . . . . . . 10 ⊢ ( O ‘𝑥) ⊆ No
31	30	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ( O ‘𝑥) ⊆ No )
32		eqidd 2767	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (( O ‘𝑥) |s ∅) = (( O ‘𝑥) |s ∅))
33	29, 31, 32	elons2d 28489	. . . . . . . 8 ⊢ (𝑥 ∈ On → (( O ‘𝑥) |s ∅) ∈ Ons)
34	33	fvresd 6908	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (( bday ↾ Ons)‘(( O ‘𝑥) |s ∅)) = ( bday ‘(( O ‘𝑥) |s ∅)))
35	28	elpw 4571	. . . . . . . . . . . . 13 ⊢ (( O ‘𝑥) ∈ 𝒫 No ↔ ( O ‘𝑥) ⊆ No )
36	30, 35	mpbir 234	. . . . . . . . . . . 12 ⊢ ( O ‘𝑥) ∈ 𝒫 No
37		nulsgts 28006	. . . . . . . . . . . 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 4354	. . . . . . . . . . . . 13 ⊢ (( O ‘𝑥) ∪ ∅) = ( O ‘𝑥)
41	40	imaeq2i 6065	. . . . . . . . . . . 12 ⊢ ( bday “ (( O ‘𝑥) ∪ ∅)) = ( bday “ ( O ‘𝑥))
42		oldbdayim 28119	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ( O ‘𝑥) → ( bday ‘𝑦) ∈ 𝑥)
43	42	rgen 3084	. . . . . . . . . . . . . 14 ⊢ ∀𝑦 ∈ ( O ‘𝑥)( bday ‘𝑦) ∈ 𝑥
44	43	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ ( O ‘𝑥)( bday ‘𝑦) ∈ 𝑥)
45	30, 5	sseqtrri 3989	. . . . . . . . . . . . . 14 ⊢ ( O ‘𝑥) ⊆ dom bday
46		funimass4 6952	. . . . . . . . . . . . . 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 3978	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ( bday “ (( O ‘𝑥) ∪ ∅)) ⊆ 𝑥)
50		cutbdaybnd 28025	. . . . . . . . . . 11 ⊢ ((( O ‘𝑥) <<s ∅ ∧ 𝑥 ∈ On ∧ ( bday “ (( O ‘𝑥) ∪ ∅)) ⊆ 𝑥) → ( bday ‘(( O ‘𝑥) |s ∅)) ⊆ 𝑥)
51	38, 39, 49, 50	mp3an2i 1495	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ( bday ‘(( O ‘𝑥) |s ∅)) ⊆ 𝑥)
52		sltssep 27997	. . . . . . . . . . . . . . . 16 ⊢ (( O ‘𝑥) <<s {𝑤} → ∀𝑦 ∈ ( O ‘𝑥)∀𝑧 ∈ {𝑤}𝑦 <s 𝑧)
53		vex 3462	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑤 ∈ V
54		breq2 5118	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝑤))
55	53, 54	ralsn 4652	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧 ∈ {𝑤}𝑦 <s 𝑧 ↔ 𝑦 <s 𝑤)
56	55	ralbii 3114	. . . . . . . . . . . . . . . 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 27947	. . . . . . . . . . . . . . . . . . 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 28131	. . . . . . . . . . . . . . . . . . . 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 5117	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑤 → (𝑦 <s 𝑤 ↔ 𝑤 <s 𝑤))
63	62	rspccv 3581	. . . . . . . . . . . . . . . . . . . 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 27982	. . . . . . . . . . . . . . . . . 18 ⊢ ( bday ‘𝑤) ∈ On
69		ontri1 6402	. . . . . . . . . . . . . . . . . 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 3160	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → ∀𝑤 ∈ No ((( O ‘𝑥) <<s {𝑤} ∧ {𝑤} <<s ∅) → 𝑥 ⊆ ( bday ‘𝑤)))
76		ssint 4934	. . . . . . . . . . . . 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 27978	. . . . . . . . . . . . . 14 ⊢ bday Fn No
78		ssrab2 4037	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)} ⊆ No
79		sseq2 3966	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ( bday ‘𝑤) → (𝑥 ⊆ 𝑧 ↔ 𝑥 ⊆ ( bday ‘𝑤)))
80	79	ralima 7242	. . . . . . . . . . . . . 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 4604	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
83	82	breq2d 5126	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → (( O ‘𝑥) <<s {𝑦} ↔ ( O ‘𝑥) <<s {𝑤}))
84	82	breq1d 5124	. . . . . . . . . . . . . . 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 3660	. . . . . . . . . . . . 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 28014	. . . . . . . . . . . 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 3981	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ ( bday ‘(( O ‘𝑥) |s ∅)))
92	51, 91	eqssd 3957	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ( bday ‘(( O ‘𝑥) |s ∅)) = 𝑥)
93	34, 92	eqtrd 2801	. . . . . . . 8 ⊢ (𝑥 ∈ On → (( bday ↾ Ons)‘(( O ‘𝑥) |s ∅)) = 𝑥)
94	27, 33, 93	rspcedvdw 3587	. . . . . . 7 ⊢ (𝑥 ∈ On → ∃𝑦 ∈ Ons (( bday ↾ Ons)‘𝑦) = 𝑥)
95		fvelrnb 6948	. . . . . . . 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 3944	. . . . 5 ⊢ On ⊆ ran ( bday ↾ Ons)
99	15, 98	eqssi 3956	. . . 4 ⊢ ran ( bday ↾ Ons) = On
100		df-fo 6549	. . . 4 ⊢ (( bday ↾ Ons):Ons–onto→On ↔ (( bday ↾ Ons) Fn Ons ∧ ran ( bday ↾ Ons) = On))
101	12, 99, 100	mpbir2an 724	. . 3 ⊢ ( bday ↾ Ons):Ons–onto→On
102		df-f1o 6550	. . 3 ⊢ (( bday ↾ Ons):Ons–1-1-onto→On ↔ (( bday ↾ Ons):Ons–1-1→On ∧ ( bday ↾ Ons):Ons–onto→On))
103	26, 101, 102	mpbir2an 724	. 2 ⊢ ( bday ↾ Ons):Ons–1-1-onto→On
104		onlts 28497	. . . . 5 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (𝑥 <s 𝑦 ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑦)))
105		fvex 6901	. . . . . 6 ⊢ ( bday ‘𝑦) ∈ V
106	105	epeli 5568	. . . . 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 5130	. . . 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 3208	. 2 ⊢ ∀𝑥 ∈ Ons ∀𝑦 ∈ Ons (𝑥 <s 𝑦 ↔ (( bday ↾ Ons)‘𝑥) E (( bday ↾ Ons)‘𝑦))
111		df-isom 6552	. 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 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 2146  ∀wral 3082  ∃wrex 3092  {crab 3419  Vcvv 3458   ∪ cun 3906   ∩ cin 3907   ⊆ wss 3908  ∅c0 4289  𝒫 cpw 4567  {csn 4594  ∩ cint 4917   class class class wbr 5114   E cep 5565  dom cdm 5666  ran crn 5667   ↾ cres 5668   “ cima 5669  Oncon0 6367  Fun wfun 6537   Fn wfn 6538  ⟶wf 6539  –1-1→wf1 6540  –onto→wfo 6541  –1-1-onto→wf1o 6542  ‘cfv 6543   Isom wiso 6544  (class class class)co 7423   No csur 27841   <s clts 27842   bday cbday 27843   <<s cslts 27987   |s ccuts 27989   O cold 28053  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-isom 6552  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-left 28060  df-right 28061  df-ons 28482

This theorem is used by:  onswe  28502  onsse  28503  addonbday  28509