Theorem zcuts 28581

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 28572	. 2 ⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
2		n0cut 28508	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))
3		n0no 28497	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
4		1no 27984	. . . . . . . . 9 ⊢ 1s ∈ No
5	4	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 1s ∈ No )
6	3, 5	subscld 28237	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → (𝐴 -s 1s ) ∈ No )
7		snelpwi 5427	. . . . . . 7 ⊢ ((𝐴 -s 1s ) ∈ No → {(𝐴 -s 1s )} ∈ 𝒫 No )
8		nulsgts 27950	. . . . . . 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 27912	. . . . . . . 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 7445	. . . . . . . . 9 ⊢ (𝐴 -s 1s ) ∈ V
13		breq1 5113	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 -s 1s ) → (𝑥 ≤s 𝑦 ↔ (𝐴 -s 1s ) ≤s 𝑦))
14	13	rexbidv 3189	. . . . . . . . 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 4648	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(𝐴 -s 1s )}∃𝑦 ∈ {(𝐴 -s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(𝐴 -s 1s )} (𝐴 -s 1s ) ≤s 𝑦)
16		breq2 5114	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 -s 1s ) → ((𝐴 -s 1s ) ≤s 𝑦 ↔ (𝐴 -s 1s ) ≤s (𝐴 -s 1s )))
17	12, 16	rexsn 4649	. . . . . . . 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 4460	. . . . . . 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 28276	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → (𝐴 -s 1s ) <s 𝐴)
23	6, 3, 22	sltssn 27944	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → {(𝐴 -s 1s )} <<s {𝐴})
24	2	sneqd 4602	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → {𝐴} = {({(𝐴 -s 1s )} |s ∅)})
25	23, 24	breqtrd 5138	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → {(𝐴 -s 1s )} <<s {({(𝐴 -s 1s )} |s ∅)})
26	3, 5	addscld 28154	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ No )
27	3	ltsp1d 28189	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 <s (𝐴 +s 1s ))
28	3, 26, 27	sltssn 27944	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → {𝐴} <<s {(𝐴 +s 1s )})
29	24, 28	eqbrtrrd 5136	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → {({(𝐴 -s 1s )} |s ∅)} <<s {(𝐴 +s 1s )})
30	9, 19, 21, 25, 29	cofcut1d 28095	. . . . 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 486	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ∈ ℕ0s) → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
33		negsfn 28197	. . . . . . . 8 ⊢ -us Fn No
34		simpl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 𝐴 ∈ No )
35	4	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 1s ∈ No )
36	34, 35	addscld 28154	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (𝐴 +s 1s ) ∈ No )
37		fnsnfv 6962	. . . . . . . 8 ⊢ (( -us Fn No ∧ (𝐴 +s 1s ) ∈ No ) → {( -us ‘(𝐴 +s 1s ))} = ( -us “ {(𝐴 +s 1s )}))
38	33, 36, 37	sylancr 598	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 +s 1s ))} = ( -us “ {(𝐴 +s 1s )}))
39		negsdi 28224	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 1s ∈ No ) → ( -us ‘(𝐴 +s 1s )) = (( -us ‘𝐴) +s ( -us ‘ 1s )))
40	34, 4, 39	sylancl 597	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 +s 1s )) = (( -us ‘𝐴) +s ( -us ‘ 1s )))
41		n0no 28497	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) ∈ No )
42	41	adantl 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) ∈ No )
43	42, 35	subsvald 28235	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (( -us ‘𝐴) -s 1s ) = (( -us ‘𝐴) +s ( -us ‘ 1s )))
44	40, 43	eqtr4d 2801	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 +s 1s )) = (( -us ‘𝐴) -s 1s ))
45	44	sneqd 4602	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 +s 1s ))} = {(( -us ‘𝐴) -s 1s )})
46	38, 45	eqtr3d 2800	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ {(𝐴 +s 1s )}) = {(( -us ‘𝐴) -s 1s )})
47	34, 35	subscld 28237	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (𝐴 -s 1s ) ∈ No )
48		fnsnfv 6962	. . . . . . . 8 ⊢ (( -us Fn No ∧ (𝐴 -s 1s ) ∈ No ) → {( -us ‘(𝐴 -s 1s ))} = ( -us “ {(𝐴 -s 1s )}))
49	33, 47, 48	sylancr 598	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 -s 1s ))} = ( -us “ {(𝐴 -s 1s )}))
50	35, 34	subsvald 28235	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( 1s -s 𝐴) = ( 1s +s ( -us ‘𝐴)))
51	34, 35	negsubsdi2d 28254	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 -s 1s )) = ( 1s -s 𝐴))
52	42, 35	addscomd 28141	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (( -us ‘𝐴) +s 1s ) = ( 1s +s ( -us ‘𝐴)))
53	50, 51, 52	3eqtr4d 2808	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 -s 1s )) = (( -us ‘𝐴) +s 1s ))
54	53	sneqd 4602	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 -s 1s ))} = {(( -us ‘𝐴) +s 1s )})
55	49, 54	eqtr3d 2800	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ {(𝐴 -s 1s )}) = {(( -us ‘𝐴) +s 1s )})
56	46, 55	oveq12d 7430	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (( -us “ {(𝐴 +s 1s )}) |s ( -us “ {(𝐴 -s 1s )})) = ({(( -us ‘𝐴) -s 1s )} |s {(( -us ‘𝐴) +s 1s )}))
57	34	ltsm1d 28276	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (𝐴 -s 1s ) <s 𝐴)
58	34	ltsp1d 28189	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 𝐴 <s (𝐴 +s 1s ))
59	47, 34, 36, 57, 58	ltstrd 27908	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (𝐴 -s 1s ) <s (𝐴 +s 1s ))
60	47, 36, 59	sltssn 27944	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {(𝐴 -s 1s )} <<s {(𝐴 +s 1s )})
61		eqidd 2764	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}) = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
62	60, 61	negsunif 28229	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})) = (( -us “ {(𝐴 +s 1s )}) |s ( -us “ {(𝐴 -s 1s )})))
63		n0cut 28508	. . . . . . 7 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) = ({(( -us ‘𝐴) -s 1s )} |s ∅))
64	4	a1i 11	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → 1s ∈ No )
65	41, 64	subscld 28237	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) -s 1s ) ∈ No )
66		snelpwi 5427	. . . . . . . . 9 ⊢ ((( -us ‘𝐴) -s 1s ) ∈ No → {(( -us ‘𝐴) -s 1s )} ∈ 𝒫 No )
67		nulsgts 27950	. . . . . . . . 9 ⊢ ({(( -us ‘𝐴) -s 1s )} ∈ 𝒫 No → {(( -us ‘𝐴) -s 1s )} <<s ∅)
68	65, 66, 67	3syl 19	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {(( -us ‘𝐴) -s 1s )} <<s ∅)
69		lesid 27912	. . . . . . . . . 10 ⊢ ((( -us ‘𝐴) -s 1s ) ∈ No → (( -us ‘𝐴) -s 1s ) ≤s (( -us ‘𝐴) -s 1s ))
70	65, 69	syl 18	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) -s 1s ) ≤s (( -us ‘𝐴) -s 1s ))
71		ovex 7445	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) -s 1s ) ∈ V
72		breq1 5113	. . . . . . . . . . . 12 ⊢ (𝑥 = (( -us ‘𝐴) -s 1s ) → (𝑥 ≤s 𝑦 ↔ (( -us ‘𝐴) -s 1s ) ≤s 𝑦))
73	72	rexbidv 3189	. . . . . . . . . . 11 ⊢ (𝑥 = (( -us ‘𝐴) -s 1s ) → (∃𝑦 ∈ {(( -us ‘𝐴) -s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(( -us ‘𝐴) -s 1s )} (( -us ‘𝐴) -s 1s ) ≤s 𝑦))
74	71, 73	ralsn 4648	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ {(( -us ‘𝐴) -s 1s )}∃𝑦 ∈ {(( -us ‘𝐴) -s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(( -us ‘𝐴) -s 1s )} (( -us ‘𝐴) -s 1s ) ≤s 𝑦)
75		breq2 5114	. . . . . . . . . . 11 ⊢ (𝑦 = (( -us ‘𝐴) -s 1s ) → ((( -us ‘𝐴) -s 1s ) ≤s 𝑦 ↔ (( -us ‘𝐴) -s 1s ) ≤s (( -us ‘𝐴) -s 1s )))
76	71, 75	rexsn 4649	. . . . . . . . . 10 ⊢ (∃𝑦 ∈ {(( -us ‘𝐴) -s 1s )} (( -us ‘𝐴) -s 1s ) ≤s 𝑦 ↔ (( -us ‘𝐴) -s 1s ) ≤s (( -us ‘𝐴) -s 1s ))
77	74, 76	bitri 278	. . . . . . . . 9 ⊢ (∀𝑥 ∈ {(( -us ‘𝐴) -s 1s )}∃𝑦 ∈ {(( -us ‘𝐴) -s 1s )}𝑥 ≤s 𝑦 ↔ (( -us ‘𝐴) -s 1s ) ≤s (( -us ‘𝐴) -s 1s ))
78	70, 77	sylibr 237	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ∀𝑥 ∈ {(( -us ‘𝐴) -s 1s )}∃𝑦 ∈ {(( -us ‘𝐴) -s 1s )}𝑥 ≤s 𝑦)
79		ral0 4460	. . . . . . . . 9 ⊢ ∀𝑥 ∈ ∅ ∃𝑦 ∈ {(( -us ‘𝐴) +s 1s )}𝑦 ≤s 𝑥
80	79	a1i 11	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ∀𝑥 ∈ ∅ ∃𝑦 ∈ {(( -us ‘𝐴) +s 1s )}𝑦 ≤s 𝑥)
81	41	ltsm1d 28276	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) -s 1s ) <s ( -us ‘𝐴))
82	65, 41, 81	sltssn 27944	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {(( -us ‘𝐴) -s 1s )} <<s {( -us ‘𝐴)})
83	63	sneqd 4602	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {( -us ‘𝐴)} = {({(( -us ‘𝐴) -s 1s )} |s ∅)})
84	82, 83	breqtrd 5138	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {(( -us ‘𝐴) -s 1s )} <<s {({(( -us ‘𝐴) -s 1s )} |s ∅)})
85	41, 64	addscld 28154	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) +s 1s ) ∈ No )
86	41	ltsp1d 28189	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) <s (( -us ‘𝐴) +s 1s ))
87	41, 85, 86	sltssn 27944	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {( -us ‘𝐴)} <<s {(( -us ‘𝐴) +s 1s )})
88	83, 87	eqbrtrrd 5136	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {({(( -us ‘𝐴) -s 1s )} |s ∅)} <<s {(( -us ‘𝐴) +s 1s )})
89	68, 78, 80, 84, 88	cofcut1d 28095	. . . . . . 7 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ({(( -us ‘𝐴) -s 1s )} |s ∅) = ({(( -us ‘𝐴) -s 1s )} |s {(( -us ‘𝐴) +s 1s )}))
90	63, 89	eqtrd 2798	. . . . . 6 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) = ({(( -us ‘𝐴) -s 1s )} |s {(( -us ‘𝐴) +s 1s )}))
91	90	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) = ({(( -us ‘𝐴) -s 1s )} |s {(( -us ‘𝐴) +s 1s )}))
92	56, 62, 91	3eqtr4rd 2809	. . . 4 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) = ( -us ‘({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})))
93	60	cutscld 27957	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}) ∈ No )
94		negs11 28223	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}) ∈ No ) → (( -us ‘𝐴) = ( -us ‘({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})) ↔ 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})))
95	93, 94	syldan 602	. . . 4 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (( -us ‘𝐴) = ( -us ‘({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})) ↔ 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})))
96	92, 95	mpbid 235	. . 3 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
97	32, 96	jaodan 972	. 2 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘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 )}))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  ∅c0 4287  𝒫 cpw 4563  {csn 4590   class class class wbr 5110   “ cima 5666   Fn wfn 6533  ‘cfv 6538  (class class class)co 7412   No csur 27785   ≤s cles 27889   <<s cslts 27931   |s ccuts 27933   1_s c1s 27980   +_s cadds 28133   -u_s cnegs 28193   -_s csubs 28194  ℕ_0scn0s 28486  ℤ_sczs 28552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-ot 4599  df-uni 4874  df-int 4914  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-nadd 8653  df-no 27788  df-lts 27789  df-bday 27790  df-les 27890  df-slts 27932  df-cuts 27934  df-0s 27981  df-1s 27982  df-made 28001  df-old 28002  df-left 28004  df-right 28005  df-norec 28112  df-norec2 28123  df-adds 28134  df-negs 28195  df-subs 28196  df-n0s 28488  df-nns 28489  df-zs 28553

This theorem is referenced by:  pw2cutp1  28635  pw2cut2  28636