Theorem zscut 28341

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

Assertion

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

Proof of Theorem zscut

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

Step	Hyp	Ref	Expression
1		elzn0s 28332	. 2 ⊢ (𝐴 ∈ ℤs ↔ (𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)))
2		n0scut 28272	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅))
3		n0sno 28262	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No )
4		1sno 27781	. . . . . . . . 9 ⊢ 1s ∈ No
5	4	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 1s ∈ No )
6	3, 5	subscld 28013	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → (𝐴 -s 1s ) ∈ No )
7		snelpwi 5389	. . . . . . 7 ⊢ ((𝐴 -s 1s ) ∈ No → {(𝐴 -s 1s )} ∈ 𝒫 No )
8		nulssgt 27749	. . . . . . 7 ⊢ ({(𝐴 -s 1s )} ∈ 𝒫 No → {(𝐴 -s 1s )} <<s ∅)
9	6, 7, 8	3syl 18	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → {(𝐴 -s 1s )} <<s ∅)
10		slerflex 27712	. . . . . . . 8 ⊢ ((𝐴 -s 1s ) ∈ No → (𝐴 -s 1s ) ≤s (𝐴 -s 1s ))
11	6, 10	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → (𝐴 -s 1s ) ≤s (𝐴 -s 1s ))
12		ovex 7388	. . . . . . . . 9 ⊢ (𝐴 -s 1s ) ∈ V
13		breq1 5098	. . . . . . . . . 10 ⊢ (𝑥 = (𝐴 -s 1s ) → (𝑥 ≤s 𝑦 ↔ (𝐴 -s 1s ) ≤s 𝑦))
14	13	rexbidv 3158	. . . . . . . . 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 4635	. . . . . . . 8 ⊢ (∀𝑥 ∈ {(𝐴 -s 1s )}∃𝑦 ∈ {(𝐴 -s 1s )}𝑥 ≤s 𝑦 ↔ ∃𝑦 ∈ {(𝐴 -s 1s )} (𝐴 -s 1s ) ≤s 𝑦)
16		breq2 5099	. . . . . . . . 9 ⊢ (𝑦 = (𝐴 -s 1s ) → ((𝐴 -s 1s ) ≤s 𝑦 ↔ (𝐴 -s 1s ) ≤s (𝐴 -s 1s )))
17	12, 16	rexsn 4636	. . . . . . . 8 ⊢ (∃𝑦 ∈ {(𝐴 -s 1s )} (𝐴 -s 1s ) ≤s 𝑦 ↔ (𝐴 -s 1s ) ≤s (𝐴 -s 1s ))
18	15, 17	bitri 275	. . . . . . 7 ⊢ (∀𝑥 ∈ {(𝐴 -s 1s )}∃𝑦 ∈ {(𝐴 -s 1s )}𝑥 ≤s 𝑦 ↔ (𝐴 -s 1s ) ≤s (𝐴 -s 1s ))
19	11, 18	sylibr 234	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → ∀𝑥 ∈ {(𝐴 -s 1s )}∃𝑦 ∈ {(𝐴 -s 1s )}𝑥 ≤s 𝑦)
20		ral0 4464	. . . . . . 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	sltm1d 28051	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → (𝐴 -s 1s ) <s 𝐴)
23	6, 3, 22	ssltsn 27743	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → {(𝐴 -s 1s )} <<s {𝐴})
24	2	sneqd 4589	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → {𝐴} = {({(𝐴 -s 1s )} |s ∅)})
25	23, 24	breqtrd 5121	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → {(𝐴 -s 1s )} <<s {({(𝐴 -s 1s )} |s ∅)})
26	3, 5	addscld 27933	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ No )
27	3	sltp1d 27968	. . . . . . . 8 ⊢ (𝐴 ∈ ℕ0s → 𝐴 <s (𝐴 +s 1s ))
28	3, 26, 27	ssltsn 27743	. . . . . . 7 ⊢ (𝐴 ∈ ℕ0s → {𝐴} <<s {(𝐴 +s 1s )})
29	24, 28	eqbrtrrd 5119	. . . . . 6 ⊢ (𝐴 ∈ ℕ0s → {({(𝐴 -s 1s )} |s ∅)} <<s {(𝐴 +s 1s )})
30	9, 19, 21, 25, 29	cofcut1d 27875	. . . . 5 ⊢ (𝐴 ∈ ℕ0s → ({(𝐴 -s 1s )} |s ∅) = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
31	2, 30	eqtrd 2768	. . . 4 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
32	31	adantl 481	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐴 ∈ ℕ0s) → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
33		negsfn 27975	. . . . . . . 8 ⊢ -us Fn No
34		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 𝐴 ∈ No )
35	4	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 1s ∈ No )
36	34, 35	addscld 27933	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (𝐴 +s 1s ) ∈ No )
37		fnsnfv 6910	. . . . . . . 8 ⊢ (( -us Fn No ∧ (𝐴 +s 1s ) ∈ No ) → {( -us ‘(𝐴 +s 1s ))} = ( -us “ {(𝐴 +s 1s )}))
38	33, 36, 37	sylancr 587	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 +s 1s ))} = ( -us “ {(𝐴 +s 1s )}))
39		negsdi 28002	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ 1s ∈ No ) → ( -us ‘(𝐴 +s 1s )) = (( -us ‘𝐴) +s ( -us ‘ 1s )))
40	34, 4, 39	sylancl 586	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 +s 1s )) = (( -us ‘𝐴) +s ( -us ‘ 1s )))
41		n0sno 28262	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) ∈ No )
42	41	adantl 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) ∈ No )
43	42, 35	subsvald 28011	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (( -us ‘𝐴) -s 1s ) = (( -us ‘𝐴) +s ( -us ‘ 1s )))
44	40, 43	eqtr4d 2771	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 +s 1s )) = (( -us ‘𝐴) -s 1s ))
45	44	sneqd 4589	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 +s 1s ))} = {(( -us ‘𝐴) -s 1s )})
46	38, 45	eqtr3d 2770	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ {(𝐴 +s 1s )}) = {(( -us ‘𝐴) -s 1s )})
47	34, 35	subscld 28013	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (𝐴 -s 1s ) ∈ No )
48		fnsnfv 6910	. . . . . . . 8 ⊢ (( -us Fn No ∧ (𝐴 -s 1s ) ∈ No ) → {( -us ‘(𝐴 -s 1s ))} = ( -us “ {(𝐴 -s 1s )}))
49	33, 47, 48	sylancr 587	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 -s 1s ))} = ( -us “ {(𝐴 -s 1s )}))
50	35, 34	subsvald 28011	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( 1s -s 𝐴) = ( 1s +s ( -us ‘𝐴)))
51	34, 35	negsubsdi2d 28030	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 -s 1s )) = ( 1s -s 𝐴))
52	42, 35	addscomd 27920	. . . . . . . . 9 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (( -us ‘𝐴) +s 1s ) = ( 1s +s ( -us ‘𝐴)))
53	50, 51, 52	3eqtr4d 2778	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘(𝐴 -s 1s )) = (( -us ‘𝐴) +s 1s ))
54	53	sneqd 4589	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {( -us ‘(𝐴 -s 1s ))} = {(( -us ‘𝐴) +s 1s )})
55	49, 54	eqtr3d 2770	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us “ {(𝐴 -s 1s )}) = {(( -us ‘𝐴) +s 1s )})
56	46, 55	oveq12d 7373	. . . . 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	sltm1d 28051	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (𝐴 -s 1s ) <s 𝐴)
58	34	sltp1d 27968	. . . . . . . 8 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 𝐴 <s (𝐴 +s 1s ))
59	47, 34, 36, 57, 58	slttrd 27708	. . . . . . 7 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → (𝐴 -s 1s ) <s (𝐴 +s 1s ))
60	47, 36, 59	ssltsn 27743	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → {(𝐴 -s 1s )} <<s {(𝐴 +s 1s )})
61		eqidd 2734	. . . . . 6 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}) = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
62	60, 61	negsunif 28007	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})) = (( -us “ {(𝐴 +s 1s )}) |s ( -us “ {(𝐴 -s 1s )})))
63		n0scut 28272	. . . . . . 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 28013	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) -s 1s ) ∈ No )
66		snelpwi 5389	. . . . . . . . 9 ⊢ ((( -us ‘𝐴) -s 1s ) ∈ No → {(( -us ‘𝐴) -s 1s )} ∈ 𝒫 No )
67		nulssgt 27749	. . . . . . . . 9 ⊢ ({(( -us ‘𝐴) -s 1s )} ∈ 𝒫 No → {(( -us ‘𝐴) -s 1s )} <<s ∅)
68	65, 66, 67	3syl 18	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {(( -us ‘𝐴) -s 1s )} <<s ∅)
69		slerflex 27712	. . . . . . . . . 10 ⊢ ((( -us ‘𝐴) -s 1s ) ∈ No → (( -us ‘𝐴) -s 1s ) ≤s (( -us ‘𝐴) -s 1s ))
70	65, 69	syl 17	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) -s 1s ) ≤s (( -us ‘𝐴) -s 1s ))
71		ovex 7388	. . . . . . . . . . 11 ⊢ (( -us ‘𝐴) -s 1s ) ∈ V
72		breq1 5098	. . . . . . . . . . . 12 ⊢ (𝑥 = (( -us ‘𝐴) -s 1s ) → (𝑥 ≤s 𝑦 ↔ (( -us ‘𝐴) -s 1s ) ≤s 𝑦))
73	72	rexbidv 3158	. . . . . . . . . . 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 4635	. . . . . . . . . 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 5099	. . . . . . . . . . 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 4636	. . . . . . . . . 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 275	. . . . . . . . 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 234	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ∀𝑥 ∈ {(( -us ‘𝐴) -s 1s )}∃𝑦 ∈ {(( -us ‘𝐴) -s 1s )}𝑥 ≤s 𝑦)
79		ral0 4464	. . . . . . . . 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	sltm1d 28051	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) -s 1s ) <s ( -us ‘𝐴))
82	65, 41, 81	ssltsn 27743	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {(( -us ‘𝐴) -s 1s )} <<s {( -us ‘𝐴)})
83	63	sneqd 4589	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {( -us ‘𝐴)} = {({(( -us ‘𝐴) -s 1s )} |s ∅)})
84	82, 83	breqtrd 5121	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {(( -us ‘𝐴) -s 1s )} <<s {({(( -us ‘𝐴) -s 1s )} |s ∅)})
85	41, 64	addscld 27933	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → (( -us ‘𝐴) +s 1s ) ∈ No )
86	41	sltp1d 27968	. . . . . . . . . 10 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) <s (( -us ‘𝐴) +s 1s ))
87	41, 85, 86	ssltsn 27743	. . . . . . . . 9 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {( -us ‘𝐴)} <<s {(( -us ‘𝐴) +s 1s )})
88	83, 87	eqbrtrrd 5119	. . . . . . . 8 ⊢ (( -us ‘𝐴) ∈ ℕ0s → {({(( -us ‘𝐴) -s 1s )} |s ∅)} <<s {(( -us ‘𝐴) +s 1s )})
89	68, 78, 80, 84, 88	cofcut1d 27875	. . . . . . 7 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ({(( -us ‘𝐴) -s 1s )} |s ∅) = ({(( -us ‘𝐴) -s 1s )} |s {(( -us ‘𝐴) +s 1s )}))
90	63, 89	eqtrd 2768	. . . . . 6 ⊢ (( -us ‘𝐴) ∈ ℕ0s → ( -us ‘𝐴) = ({(( -us ‘𝐴) -s 1s )} |s {(( -us ‘𝐴) +s 1s )}))
91	90	adantl 481	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) = ({(( -us ‘𝐴) -s 1s )} |s {(( -us ‘𝐴) +s 1s )}))
92	56, 62, 91	3eqtr4rd 2779	. . . 4 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ( -us ‘𝐴) = ( -us ‘({(𝐴 -s 1s )} |s {(𝐴 +s 1s )})))
93	60	scutcld 27754	. . . . 5 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}) ∈ No )
94		negs11 28001	. . . . 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 591	. . . 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 232	. . 3 ⊢ ((𝐴 ∈ No ∧ ( -us ‘𝐴) ∈ ℕ0s) → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
97	32, 96	jaodan 959	. 2 ⊢ ((𝐴 ∈ No ∧ (𝐴 ∈ ℕ0s ∨ ( -us ‘𝐴) ∈ ℕ0s)) → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))
98	1, 97	sylbi 217	1 ⊢ (𝐴 ∈ ℤs → 𝐴 = ({(𝐴 -s 1s )} |s {(𝐴 +s 1s )}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113  ∀wral 3049  ∃wrex 3058  ∅c0 4284  𝒫 cpw 4551  {csn 4577   class class class wbr 5095   “ cima 5624   Fn wfn 6484  ‘cfv 6489  (class class class)co 7355   No csur 27588   ≤s csle 27693   <<s csslt 27730   |s cscut 27732   1_s c1s 27777   +_s cadds 27912   -u_s cnegs 27971   -_s csubs 27972  ℕ_0scnn0s 28252  ℤ_sczs 28312

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 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4285  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 27591  df-slt 27592  df-bday 27593  df-sle 27694  df-sslt 27731  df-scut 27733  df-0s 27778  df-1s 27779  df-made 27798  df-old 27799  df-left 27801  df-right 27802  df-norec 27891  df-norec2 27902  df-adds 27913  df-negs 27973  df-subs 27974  df-n0s 28254  df-nns 28255  df-zs 28313

This theorem is referenced by:  pw2cutp1  28391  pw2cut2  28392  zs12bday  28404