Theorem n0subs 28806

Description: Subtraction of non-negative surreal integers. (Contributed by Scott Fenton, 26-May-2025.)

Assertion

Ref	Expression
n0subs	⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ (𝑁 −s 𝑀) ∈ ℕ0s))

Proof of Theorem n0subs

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

Step	Hyp	Ref	Expression
1		breq2 5112	. . . . . 6 ⊢ (𝑥 = 0s → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 0s ))
2		oveq1 7458	. . . . . . 7 ⊢ (𝑥 = 0s → (𝑥 −s 𝑧) = ( 0s −s 𝑧))
3	2	eleq1d 2849	. . . . . 6 ⊢ (𝑥 = 0s → ((𝑥 −s 𝑧) ∈ ℕ0s ↔ ( 0s −s 𝑧) ∈ ℕ0s))
4	1, 3	imbi12d 347	. . . . 5 ⊢ (𝑥 = 0s → ((𝑧 ≤s 𝑥 → (𝑥 −s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s 0s → ( 0s −s 𝑧) ∈ ℕ0s)))
5	4	ralbidv 3189	. . . 4 ⊢ (𝑥 = 0s → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 −s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 0s → ( 0s −s 𝑧) ∈ ℕ0s)))
6		breq2 5112	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 𝑦))
7		oveq1 7458	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 −s 𝑧) = (𝑦 −s 𝑧))
8	7	eleq1d 2849	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 −s 𝑧) ∈ ℕ0s ↔ (𝑦 −s 𝑧) ∈ ℕ0s))
9	6, 8	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑧 ≤s 𝑥 → (𝑥 −s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s 𝑦 → (𝑦 −s 𝑧) ∈ ℕ0s)))
10	9	ralbidv 3189	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 −s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑦 → (𝑦 −s 𝑧) ∈ ℕ0s)))
11		breq2 5112	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s (𝑦 +s 1s )))
12		oveq1 7458	. . . . . . 7 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 −s 𝑧) = ((𝑦 +s 1s ) −s 𝑧))
13	12	eleq1d 2849	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 −s 𝑧) ∈ ℕ0s ↔ ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s))
14	11, 13	imbi12d 347	. . . . 5 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑧 ≤s 𝑥 → (𝑥 −s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s)))
15	14	ralbidv 3189	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 −s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s)))
16		breq2 5112	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 𝑁))
17		oveq1 7458	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 −s 𝑧) = (𝑁 −s 𝑧))
18	17	eleq1d 2849	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝑥 −s 𝑧) ∈ ℕ0s ↔ (𝑁 −s 𝑧) ∈ ℕ0s))
19	16, 18	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑧 ≤s 𝑥 → (𝑥 −s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s 𝑁 → (𝑁 −s 𝑧) ∈ ℕ0s)))
20	19	ralbidv 3189	. . . 4 ⊢ (𝑥 = 𝑁 → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 −s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 −s 𝑧) ∈ ℕ0s)))
21		n0sge0 28781	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → 0s ≤s 𝑧)
22	21	biantrud 541	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
23		n0no 28766	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → 𝑧 ∈ No)
24		0no 28252	. . . . . . . 8 ⊢ 0s ∈ No
25		lestri3 28169	. . . . . . . 8 ⊢ ((𝑧 ∈ No ∧ 0s ∈ No) → (𝑧 = 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
26	23, 24, 25	sylancl 598	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → (𝑧 = 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
27	22, 26	bitr4d 285	. . . . . 6 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s ↔ 𝑧 = 0s ))
28		oveq2 7459	. . . . . . 7 ⊢ (𝑧 = 0s → ( 0s −s 𝑧) = ( 0s −s 0s ))
29		subsid 28512	. . . . . . . . 9 ⊢ ( 0s ∈ No → ( 0s −s 0s ) = 0s )
30	24, 29	ax-mp 5	. . . . . . . 8 ⊢ ( 0s −s 0s ) = 0s
31		0n0s 28772	. . . . . . . 8 ⊢ 0s ∈ ℕ0s
32	30, 31	eqeltri 2860	. . . . . . 7 ⊢ ( 0s −s 0s ) ∈ ℕ0s
33	28, 32	eqeltrdi 2872	. . . . . 6 ⊢ (𝑧 = 0s → ( 0s −s 𝑧) ∈ ℕ0s)
34	27, 33	biimtrdi 256	. . . . 5 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s → ( 0s −s 𝑧) ∈ ℕ0s))
35	34	rgen 3082	. . . 4 ⊢ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 0s → ( 0s −s 𝑧) ∈ ℕ0s)
36		breq1 5111	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ≤s 𝑦 ↔ 𝑥 ≤s 𝑦))
37		oveq2 7459	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 −s 𝑧) = (𝑦 −s 𝑥))
38	37	eleq1d 2849	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑦 −s 𝑧) ∈ ℕ0s ↔ (𝑦 −s 𝑥) ∈ ℕ0s))
39	36, 38	imbi12d 347	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑧 ≤s 𝑦 → (𝑦 −s 𝑧) ∈ ℕ0s) ↔ (𝑥 ≤s 𝑦 → (𝑦 −s 𝑥) ∈ ℕ0s)))
40	39	cbvralvw 3244	. . . . 5 ⊢ (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑦 → (𝑦 −s 𝑧) ∈ ℕ0s) ↔ ∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 −s 𝑥) ∈ ℕ0s))
41		n0no 28766	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No)
42		peano2no 28427	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ No → (𝑦 +s 1s ) ∈ No)
43		subsid1 28511	. . . . . . . . . . . 12 ⊢ ((𝑦 +s 1s ) ∈ No → ((𝑦 +s 1s ) −s 0s ) = (𝑦 +s 1s ))
44	41, 42, 43	3syl 19	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) −s 0s ) = (𝑦 +s 1s ))
45		peano2n0s 28773	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
46	44, 45	eqeltrd 2864	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) −s 0s ) ∈ ℕ0s)
47		oveq2 7459	. . . . . . . . . . 11 ⊢ (𝑧 = 0s → ((𝑦 +s 1s ) −s 𝑧) = ((𝑦 +s 1s ) −s 0s ))
48	47	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑧 = 0s → (((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s ↔ ((𝑦 +s 1s ) −s 0s ) ∈ ℕ0s))
49	46, 48	syl5ibrcom 250	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → (𝑧 = 0s → ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s))
50	49	2a1dd 52	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ0s → (𝑧 = 0s → (∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 −s 𝑥) ∈ ℕ0s) → (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s))))
51	50	adantr 486	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑧 = 0s → (∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 −s 𝑥) ∈ ℕ0s) → (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s))))
52		breq1 5111	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 −s 1s ) → (𝑥 ≤s 𝑦 ↔ (𝑧 −s 1s ) ≤s 𝑦))
53		oveq2 7459	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑧 −s 1s ) → (𝑦 −s 𝑥) = (𝑦 −s (𝑧 −s 1s )))
54	53	eleq1d 2849	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 −s 1s ) → ((𝑦 −s 𝑥) ∈ ℕ0s ↔ (𝑦 −s (𝑧 −s 1s )) ∈ ℕ0s))
55	52, 54	imbi12d 347	. . . . . . . . 9 ⊢ (𝑥 = (𝑧 −s 1s ) → ((𝑥 ≤s 𝑦 → (𝑦 −s 𝑥) ∈ ℕ0s) ↔ ((𝑧 −s 1s ) ≤s 𝑦 → (𝑦 −s (𝑧 −s 1s )) ∈ ℕ0s)))
56	55	rspcv 3576	. . . . . . . 8 ⊢ ((𝑧 −s 1s ) ∈ ℕ0s → (∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 −s 𝑥) ∈ ℕ0s) → ((𝑧 −s 1s ) ≤s 𝑦 → (𝑦 −s (𝑧 −s 1s )) ∈ ℕ0s)))
57	23	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → 𝑧 ∈ No)
58		1no 28253	. . . . . . . . . . . 12 ⊢ 1s ∈ No
59	58	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → 1s ∈ No)
60	41	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → 𝑦 ∈ No)
61	57, 59, 60	lesubaddsd 28536	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑧 −s 1s ) ≤s 𝑦 ↔ 𝑧 ≤s (𝑦 +s 1s )))
62	60, 57, 59	subsubs2d 28538	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 −s (𝑧 −s 1s )) = (𝑦 +s ( 1s −s 𝑧)))
63	60, 59, 57	addsubsassd 28524	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑦 +s 1s ) −s 𝑧) = (𝑦 +s ( 1s −s 𝑧)))
64	62, 63	eqtr4d 2801	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 −s (𝑧 −s 1s )) = ((𝑦 +s 1s ) −s 𝑧))
65	64	eleq1d 2849	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑦 −s (𝑧 −s 1s )) ∈ ℕ0s ↔ ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s))
66	61, 65	imbi12d 347	. . . . . . . . 9 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (((𝑧 −s 1s ) ≤s 𝑦 → (𝑦 −s (𝑧 −s 1s )) ∈ ℕ0s) ↔ (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s)))
67	66	biimpd 232	. . . . . . . 8 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (((𝑧 −s 1s ) ≤s 𝑦 → (𝑦 −s (𝑧 −s 1s )) ∈ ℕ0s) → (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s)))
68	56, 67	syl9r 79	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑧 −s 1s ) ∈ ℕ0s → (∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 −s 𝑥) ∈ ℕ0s) → (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s))))
69		n0s0m1 28805	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → (𝑧 = 0s ∨ (𝑧 −s 1s ) ∈ ℕ0s))
70	69	adantl 487	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑧 = 0s ∨ (𝑧 −s 1s ) ∈ ℕ0s))
71	51, 68, 70	mpjaod 874	. . . . . 6 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 −s 𝑥) ∈ ℕ0s) → (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s)))
72	71	ralrimdva 3166	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → (∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 −s 𝑥) ∈ ℕ0s) → ∀𝑧 ∈ ℕ0s (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s)))
73	40, 72	biimtrid 245	. . . 4 ⊢ (𝑦 ∈ ℕ0s → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑦 → (𝑦 −s 𝑧) ∈ ℕ0s) → ∀𝑧 ∈ ℕ0s (𝑧 ≤s (𝑦 +s 1s ) → ((𝑦 +s 1s ) −s 𝑧) ∈ ℕ0s)))
74	5, 10, 15, 20, 35, 73	n0sind 28776	. . 3 ⊢ (𝑁 ∈ ℕ0s → ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 −s 𝑧) ∈ ℕ0s))
75		breq1 5111	. . . . 5 ⊢ (𝑧 = 𝑀 → (𝑧 ≤s 𝑁 ↔ 𝑀 ≤s 𝑁))
76		oveq2 7459	. . . . . 6 ⊢ (𝑧 = 𝑀 → (𝑁 −s 𝑧) = (𝑁 −s 𝑀))
77	76	eleq1d 2849	. . . . 5 ⊢ (𝑧 = 𝑀 → ((𝑁 −s 𝑧) ∈ ℕ0s ↔ (𝑁 −s 𝑀) ∈ ℕ0s))
78	75, 77	imbi12d 347	. . . 4 ⊢ (𝑧 = 𝑀 → ((𝑧 ≤s 𝑁 → (𝑁 −s 𝑧) ∈ ℕ0s) ↔ (𝑀 ≤s 𝑁 → (𝑁 −s 𝑀) ∈ ℕ0s)))
79	78	rspcva 3578	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 −s 𝑧) ∈ ℕ0s)) → (𝑀 ≤s 𝑁 → (𝑁 −s 𝑀) ∈ ℕ0s))
80	74, 79	sylan2 605	. 2 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 → (𝑁 −s 𝑀) ∈ ℕ0s))
81		n0sge0 28781	. . 3 ⊢ ((𝑁 −s 𝑀) ∈ ℕ0s → 0s ≤s (𝑁 −s 𝑀))
82		n0no 28766	. . . . 5 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ No)
83	82	adantl 487	. . . 4 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑁 ∈ No)
84		n0no 28766	. . . . 5 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No)
85	84	adantr 486	. . . 4 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑀 ∈ No)
86	83, 85	subsge0d 28543	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ( 0s ≤s (𝑁 −s 𝑀) ↔ 𝑀 ≤s 𝑁))
87	81, 86	imbitrid 247	. 2 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑁 −s 𝑀) ∈ ℕ0s → 𝑀 ≤s 𝑁))
88	80, 87	impbid 215	1 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ (𝑁 −s 𝑀) ∈ ℕ0s))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080   class class class wbr 5108  (class class class)co 7451  Nocsur 28054   ≤s cles 28158   0_s c0s 28248   1_s c1s 28249   +_s cadds 28402   −_s csubs 28463  ℕ_0scn0s 28755

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-nadd 8713  df-no 28057  df-lts 28058  df-bday 28059  df-les 28159  df-slts 28201  df-cuts 28203  df-0s 28250  df-1s 28251  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec 28381  df-norec2 28392  df-adds 28403  df-negs 28464  df-subs 28465  df-n0s 28757

This theorem is used by:  n0subs2  28807  elzn0s  28841  eln0zs  28843  bdayfinbndlem1  28910