Theorem onsiso 28209

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
onsiso	⊢ ( bday ↾ Ons) Isom <s , E (Ons, On)

Proof of Theorem onsiso

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

Step	Hyp	Ref	Expression
1		bdayfun 27717	. . . . . . 7 ⊢ Fun bday
2		funres 6542	. . . . . . 7 ⊢ (Fun bday → Fun ( bday ↾ Ons))
3	1, 2	ax-mp 5	. . . . . 6 ⊢ Fun ( bday ↾ Ons)
4		dmres 5972	. . . . . . 7 ⊢ dom ( bday ↾ Ons) = (Ons ∩ dom bday )
5		bdaydm 27719	. . . . . . . 8 ⊢ dom bday = No
6	5	ineq2i 4176	. . . . . . 7 ⊢ (Ons ∩ dom bday ) = (Ons ∩ No )
7		onssno 28195	. . . . . . . 8 ⊢ Ons ⊆ No
8		dfss2 3929	. . . . . . . 8 ⊢ (Ons ⊆ No ↔ (Ons ∩ No ) = Ons)
9	7, 8	mpbi 230	. . . . . . 7 ⊢ (Ons ∩ No ) = Ons
10	4, 6, 9	3eqtri 2756	. . . . . 6 ⊢ dom ( bday ↾ Ons) = Ons
11		df-fn 6502	. . . . . 6 ⊢ (( bday ↾ Ons) Fn Ons ↔ (Fun ( bday ↾ Ons) ∧ dom ( bday ↾ Ons) = Ons))
12	3, 10, 11	mpbir2an 711	. . . . 5 ⊢ ( bday ↾ Ons) Fn Ons
13		rnresss 5977	. . . . . 6 ⊢ ran ( bday ↾ Ons) ⊆ ran bday
14		bdayrn 27720	. . . . . 6 ⊢ ran bday = On
15	13, 14	sseqtri 3992	. . . . 5 ⊢ ran ( bday ↾ Ons) ⊆ On
16		df-f 6503	. . . . 5 ⊢ (( bday ↾ Ons):Ons⟶On ↔ (( bday ↾ Ons) Fn Ons ∧ ran ( bday ↾ Ons) ⊆ On))
17	12, 15, 16	mpbir2an 711	. . . 4 ⊢ ( bday ↾ Ons):Ons⟶On
18		fvres 6859	. . . . . . 7 ⊢ (𝑥 ∈ Ons → (( bday ↾ Ons)‘𝑥) = ( bday ‘𝑥))
19		fvres 6859	. . . . . . 7 ⊢ (𝑦 ∈ Ons → (( bday ↾ Ons)‘𝑦) = ( bday ‘𝑦))
20	18, 19	eqeqan12d 2743	. . . . . 6 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → ((( bday ↾ Ons)‘𝑥) = (( bday ↾ Ons)‘𝑦) ↔ ( bday ‘𝑥) = ( bday ‘𝑦)))
21		bday11on 28206	. . . . . . 7 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons ∧ ( bday ‘𝑥) = ( bday ‘𝑦)) → 𝑥 = 𝑦)
22	21	3expia 1121	. . . . . 6 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (( bday ‘𝑥) = ( bday ‘𝑦) → 𝑥 = 𝑦))
23	20, 22	sylbid 240	. . . . 5 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → ((( bday ↾ Ons)‘𝑥) = (( bday ↾ Ons)‘𝑦) → 𝑥 = 𝑦))
24	23	rgen2 3175	. . . 4 ⊢ ∀𝑥 ∈ Ons ∀𝑦 ∈ Ons ((( bday ↾ Ons)‘𝑥) = (( bday ↾ Ons)‘𝑦) → 𝑥 = 𝑦)
25		dff13 7211	. . . 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 711	. . 3 ⊢ ( bday ↾ Ons):Ons–1-1→On
27		fveqeq2 6849	. . . . . . . 8 ⊢ (𝑦 = (( O ‘𝑥) |s ∅) → ((( bday ↾ Ons)‘𝑦) = 𝑥 ↔ (( bday ↾ Ons)‘(( O ‘𝑥) |s ∅)) = 𝑥))
28		fvex 6853	. . . . . . . . . 10 ⊢ ( O ‘𝑥) ∈ V
29	28	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ( O ‘𝑥) ∈ V)
30		oldssno 27806	. . . . . . . . . 10 ⊢ ( O ‘𝑥) ⊆ No
31	30	a1i 11	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ( O ‘𝑥) ⊆ No )
32		eqidd 2730	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (( O ‘𝑥) |s ∅) = (( O ‘𝑥) |s ∅))
33	29, 31, 32	elons2d 28200	. . . . . . . 8 ⊢ (𝑥 ∈ On → (( O ‘𝑥) |s ∅) ∈ Ons)
34	33	fvresd 6860	. . . . . . . . 9 ⊢ (𝑥 ∈ On → (( bday ↾ Ons)‘(( O ‘𝑥) |s ∅)) = ( bday ‘(( O ‘𝑥) |s ∅)))
35	28	elpw 4563	. . . . . . . . . . . . 13 ⊢ (( O ‘𝑥) ∈ 𝒫 No ↔ ( O ‘𝑥) ⊆ No )
36	30, 35	mpbir 231	. . . . . . . . . . . 12 ⊢ ( O ‘𝑥) ∈ 𝒫 No
37		nulssgt 27744	. . . . . . . . . . . 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 4353	. . . . . . . . . . . . 13 ⊢ (( O ‘𝑥) ∪ ∅) = ( O ‘𝑥)
41	40	imaeq2i 6018	. . . . . . . . . . . 12 ⊢ ( bday “ (( O ‘𝑥) ∪ ∅)) = ( bday “ ( O ‘𝑥))
42		oldbdayim 27838	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ( O ‘𝑥) → ( bday ‘𝑦) ∈ 𝑥)
43	42	rgen 3046	. . . . . . . . . . . . . 14 ⊢ ∀𝑦 ∈ ( O ‘𝑥)( bday ‘𝑦) ∈ 𝑥
44	43	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ ( O ‘𝑥)( bday ‘𝑦) ∈ 𝑥)
45	30, 5	sseqtrri 3993	. . . . . . . . . . . . . 14 ⊢ ( O ‘𝑥) ⊆ dom bday
46		funimass4 6907	. . . . . . . . . . . . . 14 ⊢ ((Fun bday ∧ ( O ‘𝑥) ⊆ dom bday ) → (( bday “ ( O ‘𝑥)) ⊆ 𝑥 ↔ ∀𝑦 ∈ ( O ‘𝑥)( bday ‘𝑦) ∈ 𝑥))
47	1, 45, 46	mp2an 692	. . . . . . . . . . . . 13 ⊢ (( bday “ ( O ‘𝑥)) ⊆ 𝑥 ↔ ∀𝑦 ∈ ( O ‘𝑥)( bday ‘𝑦) ∈ 𝑥)
48	44, 47	sylibr 234	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → ( bday “ ( O ‘𝑥)) ⊆ 𝑥)
49	41, 48	eqsstrid 3982	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → ( bday “ (( O ‘𝑥) ∪ ∅)) ⊆ 𝑥)
50		scutbdaybnd 27761	. . . . . . . . . . 11 ⊢ ((( O ‘𝑥) <<s ∅ ∧ 𝑥 ∈ On ∧ ( bday “ (( O ‘𝑥) ∪ ∅)) ⊆ 𝑥) → ( bday ‘(( O ‘𝑥) |s ∅)) ⊆ 𝑥)
51	38, 39, 49, 50	mp3an2i 1468	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → ( bday ‘(( O ‘𝑥) |s ∅)) ⊆ 𝑥)
52		ssltsep 27736	. . . . . . . . . . . . . . . 16 ⊢ (( O ‘𝑥) <<s {𝑤} → ∀𝑦 ∈ ( O ‘𝑥)∀𝑧 ∈ {𝑤}𝑦 <s 𝑧)
53		vex 3448	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑤 ∈ V
54		breq2 5106	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑤 → (𝑦 <s 𝑧 ↔ 𝑦 <s 𝑤))
55	53, 54	ralsn 4641	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑧 ∈ {𝑤}𝑦 <s 𝑧 ↔ 𝑦 <s 𝑤)
56	55	ralbii 3075	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ ( O ‘𝑥)∀𝑧 ∈ {𝑤}𝑦 <s 𝑧 ↔ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤)
57	52, 56	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (( O ‘𝑥) <<s {𝑤} → ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤)
58		sltirr 27691	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 ∈ No → ¬ 𝑤 <s 𝑤)
59	58	3ad2ant2 1134	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → ¬ 𝑤 <s 𝑤)
60		oldbday 27850	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ) → (𝑤 ∈ ( O ‘𝑥) ↔ ( bday ‘𝑤) ∈ 𝑥))
61	60	3adant3 1132	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → (𝑤 ∈ ( O ‘𝑥) ↔ ( bday ‘𝑤) ∈ 𝑥))
62		breq1 5105	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑤 → (𝑦 <s 𝑤 ↔ 𝑤 <s 𝑤))
63	62	rspccv 3582	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤 → (𝑤 ∈ ( O ‘𝑥) → 𝑤 <s 𝑤))
64	63	3ad2ant3 1135	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → (𝑤 ∈ ( O ‘𝑥) → 𝑤 <s 𝑤))
65	61, 64	sylbird 260	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → (( bday ‘𝑤) ∈ 𝑥 → 𝑤 <s 𝑤))
66	59, 65	mtod 198	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → ¬ ( bday ‘𝑤) ∈ 𝑥)
67		simp1 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → 𝑥 ∈ On)
68		bdayelon 27721	. . . . . . . . . . . . . . . . . 18 ⊢ ( bday ‘𝑤) ∈ On
69		ontri1 6354	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ ( bday ‘𝑤) ∈ On) → (𝑥 ⊆ ( bday ‘𝑤) ↔ ¬ ( bday ‘𝑤) ∈ 𝑥))
70	67, 68, 69	sylancl 586	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → (𝑥 ⊆ ( bday ‘𝑤) ↔ ¬ ( bday ‘𝑤) ∈ 𝑥))
71	66, 70	mpbird 257	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ∧ ∀𝑦 ∈ ( O ‘𝑥)𝑦 <s 𝑤) → 𝑥 ⊆ ( bday ‘𝑤))
72	71	3expia 1121	. . . . . . . . . . . . . . 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 491	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ On ∧ 𝑤 ∈ No ) → ((( O ‘𝑥) <<s {𝑤} ∧ {𝑤} <<s ∅) → 𝑥 ⊆ ( bday ‘𝑤)))
75	74	ralrimiva 3125	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → ∀𝑤 ∈ No ((( O ‘𝑥) <<s {𝑤} ∧ {𝑤} <<s ∅) → 𝑥 ⊆ ( bday ‘𝑤)))
76		ssint 4924	. . . . . . . . . . . . 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 27718	. . . . . . . . . . . . . 14 ⊢ bday Fn No
78		ssrab2 4039	. . . . . . . . . . . . . 14 ⊢ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)} ⊆ No
79		sseq2 3970	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ( bday ‘𝑤) → (𝑥 ⊆ 𝑧 ↔ 𝑥 ⊆ ( bday ‘𝑤)))
80	79	ralima 7193	. . . . . . . . . . . . . 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 692	. . . . . . . . . . . . 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 4595	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → {𝑦} = {𝑤})
83	82	breq2d 5114	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → (( O ‘𝑥) <<s {𝑦} ↔ ( O ‘𝑥) <<s {𝑤}))
84	82	breq1d 5112	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ({𝑦} <<s ∅ ↔ {𝑤} <<s ∅))
85	83, 84	anbi12d 632	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑤 → ((( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅) ↔ (( O ‘𝑥) <<s {𝑤} ∧ {𝑤} <<s ∅)))
86	85	ralrab 3662	. . . . . . . . . . . . 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 297	. . . . . . . . . . . 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 234	. . . . . . . . . . 11 ⊢ (𝑥 ∈ On → 𝑥 ⊆ ∩ ( bday “ {𝑦 ∈ No ∣ (( O ‘𝑥) <<s {𝑦} ∧ {𝑦} <<s ∅)}))
89		scutbday 27750	. . . . . . . . . . . 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 3985	. . . . . . . . . 10 ⊢ (𝑥 ∈ On → 𝑥 ⊆ ( bday ‘(( O ‘𝑥) |s ∅)))
92	51, 91	eqssd 3961	. . . . . . . . 9 ⊢ (𝑥 ∈ On → ( bday ‘(( O ‘𝑥) |s ∅)) = 𝑥)
93	34, 92	eqtrd 2764	. . . . . . . 8 ⊢ (𝑥 ∈ On → (( bday ↾ Ons)‘(( O ‘𝑥) |s ∅)) = 𝑥)
94	27, 33, 93	rspcedvdw 3588	. . . . . . 7 ⊢ (𝑥 ∈ On → ∃𝑦 ∈ Ons (( bday ↾ Ons)‘𝑦) = 𝑥)
95		fvelrnb 6903	. . . . . . . 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 234	. . . . . 6 ⊢ (𝑥 ∈ On → 𝑥 ∈ ran ( bday ↾ Ons))
98	97	ssriv 3947	. . . . 5 ⊢ On ⊆ ran ( bday ↾ Ons)
99	15, 98	eqssi 3960	. . . 4 ⊢ ran ( bday ↾ Ons) = On
100		df-fo 6505	. . . 4 ⊢ (( bday ↾ Ons):Ons–onto→On ↔ (( bday ↾ Ons) Fn Ons ∧ ran ( bday ↾ Ons) = On))
101	12, 99, 100	mpbir2an 711	. . 3 ⊢ ( bday ↾ Ons):Ons–onto→On
102		df-f1o 6506	. . 3 ⊢ (( bday ↾ Ons):Ons–1-1-onto→On ↔ (( bday ↾ Ons):Ons–1-1→On ∧ ( bday ↾ Ons):Ons–onto→On))
103	26, 101, 102	mpbir2an 711	. 2 ⊢ ( bday ↾ Ons):Ons–1-1-onto→On
104		onslt 28208	. . . . 5 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (𝑥 <s 𝑦 ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑦)))
105		fvex 6853	. . . . . 6 ⊢ ( bday ‘𝑦) ∈ V
106	105	epeli 5533	. . . . 5 ⊢ (( bday ‘𝑥) E ( bday ‘𝑦) ↔ ( bday ‘𝑥) ∈ ( bday ‘𝑦))
107	104, 106	bitr4di 289	. . . 4 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (𝑥 <s 𝑦 ↔ ( bday ‘𝑥) E ( bday ‘𝑦)))
108	18, 19	breqan12d 5118	. . . 4 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → ((( bday ↾ Ons)‘𝑥) E (( bday ↾ Ons)‘𝑦) ↔ ( bday ‘𝑥) E ( bday ‘𝑦)))
109	107, 108	bitr4d 282	. . 3 ⊢ ((𝑥 ∈ Ons ∧ 𝑦 ∈ Ons) → (𝑥 <s 𝑦 ↔ (( bday ↾ Ons)‘𝑥) E (( bday ↾ Ons)‘𝑦)))
110	109	rgen2 3175	. 2 ⊢ ∀𝑥 ∈ Ons ∀𝑦 ∈ Ons (𝑥 <s 𝑦 ↔ (( bday ↾ Ons)‘𝑥) E (( bday ↾ Ons)‘𝑦))
111		df-isom 6508	. 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 711	1 ⊢ ( bday ↾ Ons) Isom <s , E (Ons, On)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  {crab 3402  Vcvv 3444   ∪ cun 3909   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  𝒫 cpw 4559  {csn 4585  ∩ cint 4906   class class class wbr 5102   E cep 5530  dom cdm 5631  ran crn 5632   ↾ cres 5633   “ cima 5634  Oncon0 6320  Fun wfun 6493   Fn wfn 6494  ⟶wf 6495  –1-1→wf1 6496  –onto→wfo 6497  –1-1-onto→wf1o 6498  ‘cfv 6499   Isom wiso 6500  (class class class)co 7369   No csur 27584   <s cslt 27585   bday cbday 27586   <<s csslt 27726   |s cscut 27728   O cold 27788  On_scons 28192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-isom 6508  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-1o 8411  df-2o 8412  df-no 27587  df-slt 27588  df-bday 27589  df-sle 27690  df-sslt 27727  df-scut 27729  df-made 27792  df-old 27793  df-left 27795  df-right 27796  df-ons 28193

This theorem is referenced by:  onswe  28210  onsse  28211