Theorem zcuts 28850

Description: A cut expression for surreal integers. (Contributed by Scott Fenton, 20-Aug-2025.)

Assertion

Ref	Expression
zcuts	⊢ (𝐴 ∈ ℤs → 𝐴 = ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )}))

Proof of Theorem zcuts

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

Step	Hyp	Ref	Expression
1		elzn0s 28841	. 2 ⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -s‘𝐴) ∈ ℕ0s)))
2		n0cut 28777	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 −s 1s )} |s ∅))
3		n0no 28766	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No)
4		1no 28253	. . . . . . . . 9 ⊢ 1s ∈ No
5	4	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 1s ∈ No)
6	3, 5	subscld 28506	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → (𝐴 −s 1s ) ∈ No)
7		snelpwi 5419	. . . . . . 7 ⊢ ((𝐴 −s 1s ) ∈ No → {(𝐴 −s 1s )} ∈ 𝒫 No)
8		nulsgts 28219	. . . . . . 7 ⊢ ({(𝐴 −s 1s )} ∈ 𝒫 No → {(𝐴 −s 1s )} <<s ∅)
9	6, 7, 8	3syl 19	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → {(𝐴 −s 1s )} <<s ∅)
10		lesid 28181	. . . . . . . 8 ⊢ ((𝐴 −s 1s ) ∈ No → (𝐴 −s 1s ) ≤s (𝐴 −s 1s ))
11	6, 10	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → (𝐴 −s 1s ) ≤s (𝐴 −s 1s ))
12		ovex 7484	. . . . . . . . 9 ⊢ (𝐴 −s 1s ) ∈ V
13		breq1 5111	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 −s 1s ) → (𝑥 ≤s 𝑦 ↔ (𝐴 −s 1s ) ≤s 𝑦))
14	13	rexbidv 3190	. . . . . . . . 9 ⊢ (𝑥 = (𝐴 −s 1s ) → (∃𝑦 ∈ {(𝐴 −s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(𝐴 −s 1s )} (𝐴 −s 1s ) ≤s 𝑦))
15	12, 14	ralsn 4643	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(𝐴 −s 1s )}∃𝑦 ∈ {(𝐴 −s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(𝐴 −s 1s )} (𝐴 −s 1s ) ≤s 𝑦)
16		breq2 5112	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 −s 1s ) → ((𝐴 −s 1s ) ≤s 𝑦 ↔ (𝐴 −s 1s ) ≤s (𝐴 −s 1s )))
17	12, 16	rexsn 4644	. . . . . . . 8 ⊢ (∃𝑦 ∈ {(𝐴 −s 1s )} (𝐴 −s 1s ) ≤s 𝑦 ↔ (𝐴 −s 1s ) ≤s (𝐴 −s 1s ))
18	15, 17	bitri 278	. . . . . . 7 ⊢ (∀𝑥 ∈ {(𝐴 −s 1s )}∃𝑦 ∈ {(𝐴 −s 1s )}𝑥 ≤s 𝑦 ↔ (𝐴 −s 1s ) ≤s (𝐴 −s 1s ))
19	11, 18	sylibr 237	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → ∀𝑥 ∈ {(𝐴 −s 1s )}∃𝑦 ∈ {(𝐴 −s 1s )}𝑥 ≤s 𝑦)
20		ral0 4457	. . . . . . 7 ⊢ ∀𝑥 ∈ ∅ ∃𝑦 ∈ {(𝐴 +s 1s )}𝑦 ≤s 𝑥
21	20	a1i 11	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → ∀𝑥 ∈ ∅ ∃𝑦 ∈ {(𝐴 +s 1s )}𝑦 ≤s 𝑥)
22	3	ltsm1d 28545	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → (𝐴 −s 1s ) <s 𝐴)
23	6, 3, 22	sltssn 28213	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → {(𝐴 −s 1s )} <<s {𝐴})
24	2	sneqd 4596	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → {𝐴} = {({(𝐴 −s 1s )} |s ∅)})
25	23, 24	breqtrd 5137	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → {(𝐴 −s 1s )} <<s {({(𝐴 −s 1s )} |s ∅)})
26	3, 5	addscld 28423	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ No)
27	3	ltsp1d 28458	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 <s (𝐴 +s 1s ))
28	3, 26, 27	sltssn 28213	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → {𝐴} <<s {(𝐴 +s 1s )})
29	24, 28	eqbrtrrd 5135	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → {({(𝐴 −s 1s )} |s ∅)} <<s {(𝐴 +s 1s )})
30	9, 19, 21, 25, 29	cofcut1d 28364	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ({(𝐴 −s 1s )} |s ∅) = ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )}))
31	2, 30	eqtrd 2798	. . . 4 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )}))
32	31	adantl 487	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ∈ ℕ0s) → 𝐴 = ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )}))
33		negsfn 28466	. . . . . . . 8 ⊢ -s Fn No
34		simpl 488	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → 𝐴 ∈ No)
35	4	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → 1s ∈ No)
36	34, 35	addscld 28423	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → (𝐴 +s 1s ) ∈ No)
37		fnsnfv 6995	. . . . . . . 8 ⊢ (( -s Fn No ∧ (𝐴 +s 1s ) ∈ No) → {( -s‘(𝐴 +s 1s ))} = ( -s “ {(𝐴 +s 1s )}))
38	33, 36, 37	sylancr 599	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → {( -s‘(𝐴 +s 1s ))} = ( -s “ {(𝐴 +s 1s )}))
39		negsdi 28493	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 1s ∈ No) → ( -s‘(𝐴 +s 1s )) = (( -s‘𝐴) +s ( -s‘ 1s )))
40	34, 4, 39	sylancl 598	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( -s‘(𝐴 +s 1s )) = (( -s‘𝐴) +s ( -s‘ 1s )))
41		n0no 28766	. . . . . . . . . . 11 ⊢ (( -s‘𝐴) ∈ ℕ0s → ( -s‘𝐴) ∈ No)
42	41	adantl 487	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( -s‘𝐴) ∈ No)
43	42, 35	subsvald 28504	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → (( -s‘𝐴) −s 1s ) = (( -s‘𝐴) +s ( -s‘ 1s )))
44	40, 43	eqtr4d 2801	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( -s‘(𝐴 +s 1s )) = (( -s‘𝐴) −s 1s ))
45	44	sneqd 4596	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → {( -s‘(𝐴 +s 1s ))} = {(( -s‘𝐴) −s 1s )})
46	38, 45	eqtr3d 2800	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( -s “ {(𝐴 +s 1s )}) = {(( -s‘𝐴) −s 1s )})
47	34, 35	subscld 28506	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → (𝐴 −s 1s ) ∈ No)
48		fnsnfv 6995	. . . . . . . 8 ⊢ (( -s Fn No ∧ (𝐴 −s 1s ) ∈ No) → {( -s‘(𝐴 −s 1s ))} = ( -s “ {(𝐴 −s 1s )}))
49	33, 47, 48	sylancr 599	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → {( -s‘(𝐴 −s 1s ))} = ( -s “ {(𝐴 −s 1s )}))
50	35, 34	subsvald 28504	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( 1s −s 𝐴) = ( 1s +s ( -s‘𝐴)))
51	34, 35	negsubsdi2d 28523	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( -s‘(𝐴 −s 1s )) = ( 1s −s 𝐴))
52	42, 35	addscomd 28410	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → (( -s‘𝐴) +s 1s ) = ( 1s +s ( -s‘𝐴)))
53	50, 51, 52	3eqtr4d 2808	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( -s‘(𝐴 −s 1s )) = (( -s‘𝐴) +s 1s ))
54	53	sneqd 4596	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → {( -s‘(𝐴 −s 1s ))} = {(( -s‘𝐴) +s 1s )})
55	49, 54	eqtr3d 2800	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( -s “ {(𝐴 −s 1s )}) = {(( -s‘𝐴) +s 1s )})
56	46, 55	oveq12d 7469	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → (( -s “ {(𝐴 +s 1s )}) |s ( -s “ {(𝐴 −s 1s )})) = ({(( -s‘𝐴) −s 1s )} |s {(( -s‘𝐴) +s 1s )}))
57	34	ltsm1d 28545	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → (𝐴 −s 1s ) <s 𝐴)
58	34	ltsp1d 28458	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → 𝐴 <s (𝐴 +s 1s ))
59	47, 34, 36, 57, 58	ltstrd 28177	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → (𝐴 −s 1s ) <s (𝐴 +s 1s ))
60	47, 36, 59	sltssn 28213	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → {(𝐴 −s 1s )} <<s {(𝐴 +s 1s )})
61		eqidd 2764	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )}) = ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )}))
62	60, 61	negsunif 28498	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( -s‘({(𝐴 −s 1s )} |s {(𝐴 +s 1s )})) = (( -s “ {(𝐴 +s 1s )}) |s ( -s “ {(𝐴 −s 1s )})))
63		n0cut 28777	. . . . . . 7 ⊢ (( -s‘𝐴) ∈ ℕ0s → ( -s‘𝐴) = ({(( -s‘𝐴) −s 1s )} |s ∅))
64	4	a1i 11	. . . . . . . . . 10 ⊢ (( -s‘𝐴) ∈ ℕ0s → 1s ∈ No)
65	41, 64	subscld 28506	. . . . . . . . 9 ⊢ (( -s‘𝐴) ∈ ℕ0s → (( -s‘𝐴) −s 1s ) ∈ No)
66		snelpwi 5419	. . . . . . . . 9 ⊢ ((( -s‘𝐴) −s 1s ) ∈ No → {(( -s‘𝐴) −s 1s )} ∈ 𝒫 No)
67		nulsgts 28219	. . . . . . . . 9 ⊢ ({(( -s‘𝐴) −s 1s )} ∈ 𝒫 No → {(( -s‘𝐴) −s 1s )} <<s ∅)
68	65, 66, 67	3syl 19	. . . . . . . 8 ⊢ (( -s‘𝐴) ∈ ℕ0s → {(( -s‘𝐴) −s 1s )} <<s ∅)
69		lesid 28181	. . . . . . . . . 10 ⊢ ((( -s‘𝐴) −s 1s ) ∈ No → (( -s‘𝐴) −s 1s ) ≤s (( -s‘𝐴) −s 1s ))
70	65, 69	syl 18	. . . . . . . . 9 ⊢ (( -s‘𝐴) ∈ ℕ0s → (( -s‘𝐴) −s 1s ) ≤s (( -s‘𝐴) −s 1s ))
71		ovex 7484	. . . . . . . . . . 11 ⊢ (( -s‘𝐴) −s 1s ) ∈ V
72		breq1 5111	. . . . . . . . . . . 12 ⊢ (𝑥 = (( -s‘𝐴) −s 1s ) → (𝑥 ≤s 𝑦 ↔ (( -s‘𝐴) −s 1s ) ≤s 𝑦))
73	72	rexbidv 3190	. . . . . . . . . . 11 ⊢ (𝑥 = (( -s‘𝐴) −s 1s ) → (∃𝑦 ∈ {(( -s‘𝐴) −s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(( -s‘𝐴) −s 1s )} (( -s‘𝐴) −s 1s ) ≤s 𝑦))
74	71, 73	ralsn 4643	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ {(( -s‘𝐴) −s 1s )}∃𝑦 ∈ {(( -s‘𝐴) −s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(( -s‘𝐴) −s 1s )} (( -s‘𝐴) −s 1s ) ≤s 𝑦)
75		breq2 5112	. . . . . . . . . . 11 ⊢ (𝑦 = (( -s‘𝐴) −s 1s ) → ((( -s‘𝐴) −s 1s ) ≤s 𝑦 ↔ (( -s‘𝐴) −s 1s ) ≤s (( -s‘𝐴) −s 1s )))
76	71, 75	rexsn 4644	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ {(( -s‘𝐴) −s 1s )} (( -s‘𝐴) −s 1s ) ≤s 𝑦 ↔ (( -s‘𝐴) −s 1s ) ≤s (( -s‘𝐴) −s 1s ))
77	74, 76	bitri 278	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {(( -s‘𝐴) −s 1s )}∃𝑦 ∈ {(( -s‘𝐴) −s 1s )}𝑥 ≤s 𝑦 ↔ (( -s‘𝐴) −s 1s ) ≤s (( -s‘𝐴) −s 1s ))
78	70, 77	sylibr 237	. . . . . . . 8 ⊢ (( -s‘𝐴) ∈ ℕ0s → ∀𝑥 ∈ {(( -s‘𝐴) −s 1s )}∃𝑦 ∈ {(( -s‘𝐴) −s 1s )}𝑥 ≤s 𝑦)
79		ral0 4457	. . . . . . . . 9 ⊢ ∀𝑥 ∈ ∅ ∃𝑦 ∈ {(( -s‘𝐴) +s 1s )}𝑦 ≤s 𝑥
80	79	a1i 11	. . . . . . . 8 ⊢ (( -s‘𝐴) ∈ ℕ0s → ∀𝑥 ∈ ∅ ∃𝑦 ∈ {(( -s‘𝐴) +s 1s )}𝑦 ≤s 𝑥)
81	41	ltsm1d 28545	. . . . . . . . . 10 ⊢ (( -s‘𝐴) ∈ ℕ0s → (( -s‘𝐴) −s 1s ) <s ( -s‘𝐴))
82	65, 41, 81	sltssn 28213	. . . . . . . . 9 ⊢ (( -s‘𝐴) ∈ ℕ0s → {(( -s‘𝐴) −s 1s )} <<s {( -s‘𝐴)})
83	63	sneqd 4596	. . . . . . . . 9 ⊢ (( -s‘𝐴) ∈ ℕ0s → {( -s‘𝐴)} = {({(( -s‘𝐴) −s 1s )} |s ∅)})
84	82, 83	breqtrd 5137	. . . . . . . 8 ⊢ (( -s‘𝐴) ∈ ℕ0s → {(( -s‘𝐴) −s 1s )} <<s {({(( -s‘𝐴) −s 1s )} |s ∅)})
85	41, 64	addscld 28423	. . . . . . . . . 10 ⊢ (( -s‘𝐴) ∈ ℕ0s → (( -s‘𝐴) +s 1s ) ∈ No)
86	41	ltsp1d 28458	. . . . . . . . . 10 ⊢ (( -s‘𝐴) ∈ ℕ0s → ( -s‘𝐴) <s (( -s‘𝐴) +s 1s ))
87	41, 85, 86	sltssn 28213	. . . . . . . . 9 ⊢ (( -s‘𝐴) ∈ ℕ0s → {( -s‘𝐴)} <<s {(( -s‘𝐴) +s 1s )})
88	83, 87	eqbrtrrd 5135	. . . . . . . 8 ⊢ (( -s‘𝐴) ∈ ℕ0s → {({(( -s‘𝐴) −s 1s )} |s ∅)} <<s {(( -s‘𝐴) +s 1s )})
89	68, 78, 80, 84, 88	cofcut1d 28364	. . . . . . 7 ⊢ (( -s‘𝐴) ∈ ℕ0s → ({(( -s‘𝐴) −s 1s )} |s ∅) = ({(( -s‘𝐴) −s 1s )} |s {(( -s‘𝐴) +s 1s )}))
90	63, 89	eqtrd 2798	. . . . . 6 ⊢ (( -s‘𝐴) ∈ ℕ0s → ( -s‘𝐴) = ({(( -s‘𝐴) −s 1s )} |s {(( -s‘𝐴) +s 1s )}))
91	90	adantl 487	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( -s‘𝐴) = ({(( -s‘𝐴) −s 1s )} |s {(( -s‘𝐴) +s 1s )}))
92	56, 62, 91	3eqtr4rd 2809	. . . 4 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ( -s‘𝐴) = ( -s‘({(𝐴 −s 1s )} |s {(𝐴 +s 1s )})))
93	60	cutscld 28226	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )}) ∈ No)
94		negs11 28492	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )}) ∈ No) → (( -s‘𝐴) = ( -s‘({(𝐴 −s 1s )} |s {(𝐴 +s 1s )})) ↔ 𝐴 = ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )})))
95	93, 94	syldan 603	. . . 4 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → (( -s‘𝐴) = ( -s‘({(𝐴 −s 1s )} |s {(𝐴 +s 1s )})) ↔ 𝐴 = ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )})))
96	92, 95	mpbid 235	. . 3 ⊢ ((𝐴 ∈ No ∧ ( -s‘𝐴) ∈ ℕ0s) → 𝐴 = ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )}))
97	32, 96	jaodan 972	. 2 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -s‘𝐴) ∈ ℕ0s)) → 𝐴 = ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )}))
98	1, 97	sylbi 220	1 ⊢ (𝐴 ∈ ℤs → 𝐴 = ({(𝐴 −s 1s )} |s {(𝐴 +s 1s )}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ∅c0 4281  𝒫 cpw 4560  {csn 4588   class class class wbr 5108   “ cima 5661   Fn wfn 6561  ‘cfv 6566  (class class class)co 7451  Nocsur 28054   ≤s cles 28158   <<s cslts 28200   |s ccuts 28202   1_s c1s 28249   +_s cadds 28402   -_scnegs 28462   −_s csubs 28463  ℕ_0scn0s 28755  ℤ_sczs 28821

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  df-nns 28758  df-zs 28822

This theorem is used by:  pw2cutp1  28904  pw2cut2  28905