Theorem n0subs 28260

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 5114	. . . . . 6 ⊢ (𝑥 = 0s → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 0s ))
2		oveq1 7397	. . . . . . 7 ⊢ (𝑥 = 0s → (𝑥 -s 𝑧) = ( 0s -s 𝑧))
3	2	eleq1d 2814	. . . . . 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 3157	. . . 4 ⊢ (𝑥 = 0s → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s)))
6		breq2 5114	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 𝑦))
7		oveq1 7397	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 -s 𝑧) = (𝑦 -s 𝑧))
8	7	eleq1d 2814	. . . . . 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 3157	. . . 4 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s)))
11		breq2 5114	. . . . . 6 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s (𝑦 +s 1s )))
12		oveq1 7397	. . . . . . 7 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 𝑧) = ((𝑦 +s 1s ) -s 𝑧))
13	12	eleq1d 2814	. . . . . 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 3157	. . . 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 5114	. . . . . 6 ⊢ (𝑥 = 𝑁 → (𝑧 ≤s 𝑥 ↔ 𝑧 ≤s 𝑁))
17		oveq1 7397	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (𝑥 -s 𝑧) = (𝑁 -s 𝑧))
18	17	eleq1d 2814	. . . . . 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 3157	. . . 4 ⊢ (𝑥 = 𝑁 → (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑥 → (𝑥 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s)))
21		n0sge0 28237	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → 0s ≤s 𝑧)
22	21	biantrud 531	. . . . . . 7 ⊢ (𝑧 ∈ ℕ0s → (𝑧 ≤s 0s ↔ (𝑧 ≤s 0s ∧ 0s ≤s 𝑧)))
23		n0sno 28223	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ0s → 𝑧 ∈ No )
24		0sno 27745	. . . . . . . 8 ⊢ 0s ∈ No
25		sletri3 27674	. . . . . . . 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 7398	. . . . . . 7 ⊢ (𝑧 = 0s → ( 0s -s 𝑧) = ( 0s -s 0s ))
29		subsid 27980	. . . . . . . . 9 ⊢ ( 0s ∈ No → ( 0s -s 0s ) = 0s )
30	24, 29	ax-mp 5	. . . . . . . 8 ⊢ ( 0s -s 0s ) = 0s
31		0n0s 28229	. . . . . . . 8 ⊢ 0s ∈ ℕ0s
32	30, 31	eqeltri 2825	. . . . . . 7 ⊢ ( 0s -s 0s ) ∈ ℕ0s
33	28, 32	eqeltrdi 2837	. . . . . 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 3047	. . . 4 ⊢ ∀𝑧 ∈ ℕ0s (𝑧 ≤s 0s → ( 0s -s 𝑧) ∈ ℕ0s)
36		breq1 5113	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧 ≤s 𝑦 ↔ 𝑥 ≤s 𝑦))
37		oveq2 7398	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (𝑦 -s 𝑧) = (𝑦 -s 𝑥))
38	37	eleq1d 2814	. . . . . . 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 3216	. . . . 5 ⊢ (∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑦 → (𝑦 -s 𝑧) ∈ ℕ0s) ↔ ∀𝑥 ∈ ℕ0s (𝑥 ≤s 𝑦 → (𝑦 -s 𝑥) ∈ ℕ0s))
41		n0sno 28223	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No )
42		peano2no 27898	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ No → (𝑦 +s 1s ) ∈ No )
43		subsid1 27979	. . . . . . . . . . . 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 28230	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
46	44, 45	eqeltrd 2829	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 0s ) ∈ ℕ0s)
47		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑧 = 0s → ((𝑦 +s 1s ) -s 𝑧) = ((𝑦 +s 1s ) -s 0s ))
48	47	eleq1d 2814	. . . . . . . . . 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 5113	. . . . . . . . . 10 ⊢ (𝑥 = (𝑧 -s 1s ) → (𝑥 ≤s 𝑦 ↔ (𝑧 -s 1s ) ≤s 𝑦))
53		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑧 -s 1s ) → (𝑦 -s 𝑥) = (𝑦 -s (𝑧 -s 1s )))
54	53	eleq1d 2814	. . . . . . . . . 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 3587	. . . . . . . 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 27746	. . . . . . . . . . . 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 28004	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑧 -s 1s ) ≤s 𝑦 ↔ 𝑧 ≤s (𝑦 +s 1s )))
62	60, 57, 59	subsubs2d 28006	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 -s (𝑧 -s 1s )) = (𝑦 +s ( 1s -s 𝑧)))
63	60, 59, 57	addsubsassd 27992	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → ((𝑦 +s 1s ) -s 𝑧) = (𝑦 +s ( 1s -s 𝑧)))
64	62, 63	eqtr4d 2768	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ0s ∧ 𝑧 ∈ ℕ0s) → (𝑦 -s (𝑧 -s 1s )) = ((𝑦 +s 1s ) -s 𝑧))
65	64	eleq1d 2814	. . . . . . . . . 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 28259	. . . . . . . 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 3134	. . . . 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 28232	. . 3 ⊢ (𝑁 ∈ ℕ0s → ∀𝑧 ∈ ℕ0s (𝑧 ≤s 𝑁 → (𝑁 -s 𝑧) ∈ ℕ0s))
75		breq1 5113	. . . . 5 ⊢ (𝑧 = 𝑀 → (𝑧 ≤s 𝑁 ↔ 𝑀 ≤s 𝑁))
76		oveq2 7398	. . . . . 6 ⊢ (𝑧 = 𝑀 → (𝑁 -s 𝑧) = (𝑁 -s 𝑀))
77	76	eleq1d 2814	. . . . 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 3589	. . 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 28237	. . 3 ⊢ ((𝑁 -s 𝑀) ∈ ℕ0s → 0s ≤s (𝑁 -s 𝑀))
82		n0sno 28223	. . . . 5 ⊢ (𝑁 ∈ ℕ0s → 𝑁 ∈ No )
83	82	adantl 481	. . . 4 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑁 ∈ No )
84		n0sno 28223	. . . . 5 ⊢ (𝑀 ∈ ℕ0s → 𝑀 ∈ No )
85	84	adantr 480	. . . 4 ⊢ ((𝑀 ∈ ℕ0s ∧ 𝑁 ∈ ℕ0s) → 𝑀 ∈ No )
86	83, 85	subsge0d 28010	. . 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 1540   ∈ wcel 2109  ∀wral 3045   class class class wbr 5110  (class class class)co 7390   No csur 27558   ≤s csle 27663   0_s c0s 27741   1_s c1s 27742   +_s cadds 27873   -_s csubs 27933  ℕ_0scnn0s 28213

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-nadd 8633  df-no 27561  df-slt 27562  df-bday 27563  df-sle 27664  df-sslt 27700  df-scut 27702  df-0s 27743  df-1s 27744  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec 27852  df-norec2 27863  df-adds 27874  df-negs 27934  df-subs 27935  df-n0s 28215

This theorem is referenced by:  n0subs2  28261  elzn0s  28293  eln0zs  28295