Theorem zcuts 28558

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 28549	. 2 ⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
2		n0cut 28485	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))
3		n0no 28474	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
4		1no 27961	. . . . . . . . 9 ⊢ 1s ∈ No
5	4	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 1s ∈ No )
6	3, 5	subscld 28214	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → (𝐴 -s 1s ) ∈ No )
7		snelpwi 5416	. . . . . . 7 ⊢ ((𝐴 -s 1s ) ∈ No → {(𝐴 -s 1s )} ∈ 𝒫 No )
8		nulsgts 27927	. . . . . . 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 27889	. . . . . . . 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 7433	. . . . . . . . 9 ⊢ (𝐴 -s 1s ) ∈ V
13		breq1 5108	. . . . . . . . . 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 4643	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(𝐴 -s 1s )}∃𝑦 ∈ {(𝐴 -s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(𝐴 -s 1s )} (𝐴 -s 1s ) ≤s 𝑦)
16		breq2 5109	. . . . . . . . 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 4455	. . . . . . 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 28253	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → (𝐴 -s 1s ) <s 𝐴)
23	6, 3, 22	sltssn 27921	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → {(𝐴 -s 1s )} <<s {𝐴})
24	2	sneqd 4597	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → {𝐴} = {({(𝐴 -s 1s )} |s ∅)})
25	23, 24	breqtrd 5131	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → {(𝐴 -s 1s )} <<s {({(𝐴 -s 1s )} |s ∅)})
26	3, 5	addscld 28131	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ No )
27	3	ltsp1d 28166	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 <s (𝐴 +s 1s ))
28	3, 26, 27	sltssn 27921	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → {𝐴} <<s {(𝐴 +s 1s )})
29	24, 28	eqbrtrrd 5129	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → {({(𝐴 -s 1s )} |s ∅)} <<s {(𝐴 +s 1s )})
30	9, 19, 21, 25, 29	cofcut1d 28072	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ({(𝐴 -s 1s )} |s ∅) = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
31	2, 30	eqtrd 2800	. . . 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 28174	. . . . . . . 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 28131	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (𝐴 +s 1s ) ∈ No )
37		fnsnfv 6950	. . . . . . . 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 28201	. . . . . . . . . 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 28474	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) ∈ No )
42	41	adantl 486	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) ∈ No )
43	42, 35	subsvald 28212	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (( -us ‘𝐴) -s 1s ) = (( -us ‘𝐴) +s ( -us ‘ 1s )))
44	40, 43	eqtr4d 2803	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 +s 1s )) = (( -us ‘𝐴) -s 1s ))
45	44	sneqd 4597	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 +s 1s ))} = {(( -us ‘𝐴) -s 1s )})
46	38, 45	eqtr3d 2802	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ {(𝐴 +s 1s )}) = {(( -us ‘𝐴) -s 1s )})
47	34, 35	subscld 28214	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (𝐴 -s 1s ) ∈ No )
48		fnsnfv 6950	. . . . . . . 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 28212	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( 1s -s 𝐴) = ( 1s +s ( -us ‘𝐴)))
51	34, 35	negsubsdi2d 28231	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 -s 1s )) = ( 1s -s 𝐴))
52	42, 35	addscomd 28118	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (( -us ‘𝐴) +s 1s ) = ( 1s +s ( -us ‘𝐴)))
53	50, 51, 52	3eqtr4d 2810	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 -s 1s )) = (( -us ‘𝐴) +s 1s ))
54	53	sneqd 4597	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 -s 1s ))} = {(( -us ‘𝐴) +s 1s )})
55	49, 54	eqtr3d 2802	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ {(𝐴 -s 1s )}) = {(( -us ‘𝐴) +s 1s )})
56	46, 55	oveq12d 7418	. . . . 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 28253	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (𝐴 -s 1s ) <s 𝐴)
58	34	ltsp1d 28166	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 𝐴 <s (𝐴 +s 1s ))
59	47, 34, 36, 57, 58	ltstrd 27885	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (𝐴 -s 1s ) <s (𝐴 +s 1s ))
60	47, 36, 59	sltssn 27921	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {(𝐴 -s 1s )} <<s {(𝐴 +s 1s )})
61		eqidd 2766	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}) = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
62	60, 61	negsunif 28206	. . . . 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 28485	. . . . . . 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 28214	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) -s 1s ) ∈ No )
66		snelpwi 5416	. . . . . . . . 9 ⊢ ((( -us ‘𝐴) -s 1s ) ∈ No → {(( -us ‘𝐴) -s 1s )} ∈ 𝒫 No )
67		nulsgts 27927	. . . . . . . . 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 27889	. . . . . . . . . 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 7433	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) -s 1s ) ∈ V
72		breq1 5108	. . . . . . . . . . . 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 4643	. . . . . . . . . 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 5109	. . . . . . . . . . 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 4644	. . . . . . . . . 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 4455	. . . . . . . . 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 28253	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) -s 1s ) <s ( -us ‘𝐴))
82	65, 41, 81	sltssn 27921	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {(( -us ‘𝐴) -s 1s )} <<s {( -us ‘𝐴)})
83	63	sneqd 4597	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {( -us ‘𝐴)} = {({(( -us ‘𝐴) -s 1s )} |s ∅)})
84	82, 83	breqtrd 5131	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {(( -us ‘𝐴) -s 1s )} <<s {({(( -us ‘𝐴) -s 1s )} |s ∅)})
85	41, 64	addscld 28131	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) +s 1s ) ∈ No )
86	41	ltsp1d 28166	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) <s (( -us ‘𝐴) +s 1s ))
87	41, 85, 86	sltssn 27921	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {( -us ‘𝐴)} <<s {(( -us ‘𝐴) +s 1s )})
88	83, 87	eqbrtrrd 5129	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {({(( -us ‘𝐴) -s 1s )} |s ∅)} <<s {(( -us ‘𝐴) +s 1s )})
89	68, 78, 80, 84, 88	cofcut1d 28072	. . . . . . 7 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ({(( -us ‘𝐴) -s 1s )} |s ∅) = ({(( -us ‘𝐴) -s 1s )} |s {(( -us ‘𝐴) +s 1s )}))
90	63, 89	eqtrd 2800	. . . . . 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 2811	. . . 4 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) = ( -us ‘({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})))
93	60	cutscld 27934	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}) ∈ No )
94		negs11 28200	. . . . 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 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  ∅c0 4288  𝒫 cpw 4558  {csn 4585   class class class wbr 5105   “ cima 5655   Fn wfn 6520  ‘cfv 6525  (class class class)co 7400   No csur 27762   ≤s cles 27866   <<s cslts 27908   |s ccuts 27910   1_s c1s 27957   +_s cadds 28110   -u_s cnegs 28170   -_s csubs 28171  ℕ_0scn0s 28463  ℤ_sczs 28529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-nadd 8640  df-no 27765  df-lts 27766  df-bday 27767  df-les 27867  df-slts 27909  df-cuts 27911  df-0s 27958  df-1s 27959  df-made 27978  df-old 27979  df-left 27981  df-right 27982  df-norec 28089  df-norec2 28100  df-adds 28111  df-negs 28172  df-subs 28173  df-n0s 28465  df-nns 28466  df-zs 28530

This theorem is referenced by:  pw2cutp1  28612  pw2cut2  28613