Theorem n0subs 28309

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 5099	. . . . . 6 ⊢ (𝑥 = 0s → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 0s ))
2		oveq1 7362	. . . . . . 7 ⊢ (𝑥 = 0s → (𝑥 -s 𝑧) = ( 0s -s 𝑧))
3	2	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = 0s → ((𝑥 -s 𝑧) ∈ ℕ0s ↔ ( 0s -s 𝑧) ∈ ℕ0s))
4	1, 3	imbi12d 344	. . . . 5 ⊢ (𝑥 = 0s → ((𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s)))
5	4	ralbidv 3156	. . . 4 ⊢ (𝑥 = 0s → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s)))
6		breq2 5099	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 𝑦))
7		oveq1 7362	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 -s 𝑧) = (𝑦 -s 𝑧))
8	7	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 -s 𝑧) ∈ ℕ0s ↔ (𝑦 -s 𝑧) ∈ ℕ0s))
9	6, 8	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s)))
10	9	ralbidv 3156	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s)))
11		breq2 5099	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s (𝑦 +s 1s )))
12		oveq1 7362	. . . . . . 7 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 𝑧) = ((𝑦 +s 1s ) -s 𝑧))
13	12	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 -s 𝑧) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s))
14	11, 13	imbi12d 344	. . . . 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 3156	. . . 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 5099	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 𝑁))
17		oveq1 7362	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 -s 𝑧) = (𝑁 -s 𝑧))
18	17	eleq1d 2818	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((𝑥 -s 𝑧) ∈ ℕ0s ↔ (𝑁 -s 𝑧) ∈ ℕ0s))
19	16, 18	imbi12d 344	. . . . 5 ⊢ (𝑥 = 𝑁 → ((𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s)))
20	19	ralbidv 3156	. . . 4 ⊢ (𝑥 = 𝑁 → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s)))
21		n0sge0 28286	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → 0s ≤s 𝑧)
22	21	biantrud 531	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
23		n0sno 28272	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → 𝑧 ∈ No )
24		0sno 27790	. . . . . . . 8 ⊢ 0s ∈ No
25		sletri3 27714	. . . . . . . 8 ⊢ ((𝑧 ∈ No ∧ 0s ∈ No ) → (𝑧 = 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
26	23, 24, 25	sylancl 586	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → (𝑧 = 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
27	22, 26	bitr4d 282	. . . . . 6 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s ↔ 𝑧 = 0s ))
28		oveq2 7363	. . . . . . 7 ⊢ (𝑧 = 0s → ( 0s -s 𝑧) = ( 0s -s 0s ))
29		subsid 28029	. . . . . . . . 9 ⊢ ( 0s ∈ No → ( 0s -s 0s ) = 0s )
30	24, 29	ax-mp 5	. . . . . . . 8 ⊢ ( 0s -s 0s ) = 0s
31		0n0s 28278	. . . . . . . 8 ⊢ 0s ∈ ℕ0s
32	30, 31	eqeltri 2829	. . . . . . 7 ⊢ ( 0s -s 0s ) ∈ ℕ0s
33	28, 32	eqeltrdi 2841	. . . . . 6 ⊢ (𝑧 = 0s → ( 0s -s 𝑧) ∈ ℕ0s)
34	27, 33	biimtrdi 253	. . . . 5 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s))
35	34	rgen 3050	. . . 4 ⊢ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s)
36		breq1 5098	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ≤s 𝑦 ↔ 𝑥 ≤s 𝑦))
37		oveq2 7363	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 -s 𝑧) = (𝑦 -s 𝑥))
38	37	eleq1d 2818	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑦 -s 𝑧) ∈ ℕ0s ↔ (𝑦 -s 𝑥) ∈ ℕ0s))
39	36, 38	imbi12d 344	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s) ↔ (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s)))
40	39	cbvralvw 3211	. . . . 5 ⊢ (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s))
41		n0sno 28272	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No )
42		peano2no 27947	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ No → (𝑦 +s 1s ) ∈ No )
43		subsid1 28028	. . . . . . . . . . . 12 ⊢ ((𝑦 +s 1s ) ∈ No → ((𝑦 +s 1s ) -s 0s ) = (𝑦 +s 1s ))
44	41, 42, 43	3syl 18	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 0s ) = (𝑦 +s 1s ))
45		peano2n0s 28279	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
46	44, 45	eqeltrd 2833	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 0s ) ∈ ℕ0s)
47		oveq2 7363	. . . . . . . . . . 11 ⊢ (𝑧 = 0s → ((𝑦 +s 1s ) -s 𝑧) = ((𝑦 +s 1s ) -s 0s ))
48	47	eleq1d 2818	. . . . . . . . . 10 ⊢ (𝑧 = 0s → (((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 0s ) ∈ ℕ0s))
49	46, 48	syl5ibrcom 247	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ0s → (𝑧 = 0s → ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s))
50	49	2a1dd 51	. . . . . . . 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 480	. . . . . . 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 5098	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 -s 1s ) → (𝑥 ≤s 𝑦 ↔ (𝑧 -s 1s ) ≤s 𝑦))
53		oveq2 7363	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑧 -s 1s ) → (𝑦 -s 𝑥) = (𝑦 -s (𝑧 -s 1s )))
54	53	eleq1d 2818	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 -s 1s ) → ((𝑦 -s 𝑥) ∈ ℕ0s ↔ (𝑦 -s (𝑧 -s 1s )) ∈ ℕ0s))
55	52, 54	imbi12d 344	. . . . . . . . 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 3569	. . . . . . . 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 481	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → 𝑧 ∈ No )
58		1sno 27791	. . . . . . . . . . . 12 ⊢ 1s ∈ No
59	58	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → 1s ∈ No )
60	41	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → 𝑦 ∈ No )
61	57, 59, 60	slesubaddd 28053	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑧 -s 1s ) ≤s 𝑦 ↔ 𝑧 ≤s (𝑦 +s 1s )))
62	60, 57, 59	subsubs2d 28055	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 -s (𝑧 -s 1s )) = (𝑦 +s ( 1s -s 𝑧)))
63	60, 59, 57	addsubsassd 28041	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑦 +s 1s ) -s 𝑧) = (𝑦 +s ( 1s -s 𝑧)))
64	62, 63	eqtr4d 2771	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 -s (𝑧 -s 1s )) = ((𝑦 +s 1s ) -s 𝑧))
65	64	eleq1d 2818	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑦 -s (𝑧 -s 1s )) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 𝑧) ∈ ℕ0s))
66	61, 65	imbi12d 344	. . . . . . . . 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 229	. . . . . . . 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 78	. . . . . . 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 28308	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → (𝑧 = 0s ∨ (𝑧 -s 1s ) ∈ ℕ0s))
70	69	adantl 481	. . . . . . 7 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑧 = 0s ∨ (𝑧 -s 1s ) ∈ ℕ0s))
71	51, 68, 70	mpjaod 860	. . . . . 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 3133	. . . . 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 242	. . . 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 28281	. . 3 ⊢ (𝑁 ∈ ℕ0s → ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s))
75		breq1 5098	. . . . 5 ⊢ (𝑧 = 𝑀 → (𝑧 ≤s 𝑁 ↔ 𝑀 ≤s 𝑁))
76		oveq2 7363	. . . . . 6 ⊢ (𝑧 = 𝑀 → (𝑁 -s 𝑧) = (𝑁 -s 𝑀))
77	76	eleq1d 2818	. . . . 5 ⊢ (𝑧 = 𝑀 → ((𝑁 -s 𝑧) ∈ ℕ0s ↔ (𝑁 -s 𝑀) ∈ ℕ0s))
78	75, 77	imbi12d 344	. . . 4 ⊢ (𝑧 = 𝑀 → ((𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s) ↔ (𝑀 ≤s 𝑁 → (𝑁 -s 𝑀) ∈ ℕ0s)))
79	78	rspcva 3571	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s)) → (𝑀 ≤s 𝑁 → (𝑁 -s 𝑀) ∈ ℕ0s))
80	74, 79	sylan2 593	. 2 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 → (𝑁 -s 𝑀) ∈ ℕ0s))
81		n0sge0 28286	. . 3 ⊢ ((𝑁 -s 𝑀) ∈ ℕ0s → 0s ≤s (𝑁 -s 𝑀))
82		n0sno 28272	. . . . 5 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ No )
83	82	adantl 481	. . . 4 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑁 ∈ No )
84		n0sno 28272	. . . . 5 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
85	84	adantr 480	. . . 4 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑀 ∈ No )
86	83, 85	subsge0d 28059	. . 3 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ( 0s ≤s (𝑁 -s 𝑀) ↔ 𝑀 ≤s 𝑁))
87	81, 86	imbitrid 244	. 2 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → ((𝑁 -s 𝑀) ∈ ℕ0s → 𝑀 ≤s 𝑁))
88	80, 87	impbid 212	1 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → (𝑀 ≤s 𝑁 ↔ (𝑁 -s 𝑀) ∈ ℕ0s))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∀wral 3048   class class class wbr 5095  (class class class)co 7355   No csur 27598   ≤s csle 27703   0_s c0s 27786   1_s c1s 27787   +_s cadds 27922   -_s csubs 27982  ℕ_0scnn0s 28262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7677

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-tp 4582  df-op 4584  df-ot 4586  df-uni 4861  df-int 4900  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-se 5575  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-pred 6256  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-om 7806  df-1st 7930  df-2nd 7931  df-frecs 8220  df-wrecs 8251  df-recs 8300  df-rdg 8338  df-1o 8394  df-2o 8395  df-nadd 8590  df-no 27601  df-slt 27602  df-bday 27603  df-sle 27704  df-sslt 27741  df-scut 27743  df-0s 27788  df-1s 27789  df-made 27808  df-old 27809  df-left 27811  df-right 27812  df-norec 27901  df-norec2 27912  df-adds 27923  df-negs 27983  df-subs 27984  df-n0s 28264

This theorem is referenced by:  n0subs2  28310  elzn0s  28342  eln0zs  28344