Theorem n0fincut 28526

Description: The simplest number greater than a finite set of non-negative surreal integers is a non-negative surreal integer. (Contributed by Scott Fenton, 5-Nov-2025.)

Assertion

Ref	Expression
n0fincut	⊢ ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s)

Proof of Theorem n0fincut

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

Step	Hyp	Ref	Expression
1		oveq1 7419	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 |s ∅) = (∅ |s ∅))
2		df-0s 27978	. . . . 5 ⊢ 0s = (∅ |s ∅)
3		0n0s 28500	. . . . 5 ⊢ 0s ∈ ℕ0s
4	2, 3	eqeltrri 2860	. . . 4 ⊢ (∅ |s ∅) ∈ ℕ0s
5	1, 4	eqeltrdi 2871	. . 3 ⊢ (𝐴 = ∅ → (𝐴 |s ∅) ∈ ℕ0s)
6	5	a1d 26	. 2 ⊢ (𝐴 = ∅ → ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s))
7		n0ssno 28491	. . . . . . . 8 ⊢ ℕ0s ⊆ No
8		sstr 3946	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℕ0s ∧ ℕ0s ⊆ No ) → 𝐴 ⊆ No )
9	7, 8	mpan2 703	. . . . . . 7 ⊢ (𝐴 ⊆ ℕ0s → 𝐴 ⊆ No )
10		ltsso 27818	. . . . . . 7 ⊢ <s Or No
11		soss 5591	. . . . . . 7 ⊢ (𝐴 ⊆ No → ( <s Or No → <s Or 𝐴))
12	9, 10, 11	mpisyl 22	. . . . . 6 ⊢ (𝐴 ⊆ ℕ0s → <s Or 𝐴)
13	12	ad2antrl 740	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → <s Or 𝐴)
14		simprr 784	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ Fin)
15		simpl 487	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → 𝐴 ≠ ∅)
16		fimax2g 9247	. . . . 5 ⊢ (( <s Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
17	13, 14, 15, 16	syl3anc 1398	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
18	9	ad2antrl 740	. . . . . . . . . 10 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → 𝐴 ⊆ No )
19	18	adantr 485	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ No )
20	19	sselda 3938	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ No )
21	18	sselda 3938	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No )
22	21	adantr 485	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ No )
23		lenlts 27894	. . . . . . . 8 ⊢ ((𝑦 ∈ No ∧ 𝑥 ∈ No ) → (𝑦 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑦))
24	20, 22, 23	syl2anc 595	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑦))
25	24	ralbidva 3186	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
26		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥) → 𝑥 ∈ 𝐴)
27		ssel2 3933	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No )
28	18, 26, 27	syl2an 607	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ No )
29		snelpwi 5427	. . . . . . . . . . 11 ⊢ (𝑥 ∈ No → {𝑥} ∈ 𝒫 No )
30		nulsgts 27947	. . . . . . . . . . 11 ⊢ ({𝑥} ∈ 𝒫 No → {𝑥} <<s ∅)
31	28, 29, 30	3syl 19	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {𝑥} <<s ∅)
32		breq2 5114	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑥 ≤s 𝑤 ↔ 𝑥 ≤s 𝑥))
33		simprl 782	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ 𝐴)
34		lesid 27909	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ No → 𝑥 ≤s 𝑥)
35	28, 34	syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ≤s 𝑥)
36	32, 33, 35	rspcedvdw 3585	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ∃𝑤 ∈ 𝐴 𝑥 ≤s 𝑤)
37		vex 3459	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
38		breq1 5113	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ≤s 𝑤 ↔ 𝑥 ≤s 𝑤))
39	38	rexbidv 3189	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (∃𝑤 ∈ 𝐴 𝑧 ≤s 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑥 ≤s 𝑤))
40	37, 39	ralsn 4648	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ {𝑥}∃𝑤 ∈ 𝐴 𝑧 ≤s 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑥 ≤s 𝑤)
41	36, 40	sylibr 237	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ∀𝑧 ∈ {𝑥}∃𝑤 ∈ 𝐴 𝑧 ≤s 𝑤)
42		ral0 4460	. . . . . . . . . . 11 ⊢ ∀𝑧 ∈ ∅ ∃𝑤 ∈ ∅ 𝑤 ≤s 𝑧
43	42	a1i 11	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ∀𝑧 ∈ ∅ ∃𝑤 ∈ ∅ 𝑤 ≤s 𝑧)
44		simplrr 789	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 ∈ Fin)
45		snex 5412	. . . . . . . . . . . 12 ⊢ {({𝑥} |s ∅)} ∈ V
46	45	a1i 11	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {({𝑥} |s ∅)} ∈ V)
47	18	adantr 485	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 ⊆ No )
48	31	cutscld 27954	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) ∈ No )
49	48	snssd 4753	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {({𝑥} |s ∅)} ⊆ No )
50	47	sselda 3938	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ No )
51	28	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ No )
52	48	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → ({𝑥} |s ∅) ∈ No )
53		breq1 5113	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝑦 ≤s 𝑥 ↔ 𝑧 ≤s 𝑥))
54		simplrr 789	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)
55		simpr 489	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
56	53, 54, 55	rspcdva 3583	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤s 𝑥)
57	51, 34	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑥 ≤s 𝑥)
58		breq2 5114	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑥 → (𝑥 ≤s 𝑧 ↔ 𝑥 ≤s 𝑥))
59	37, 58	rexsn 4649	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑧 ∈ {𝑥}𝑥 ≤s 𝑧 ↔ 𝑥 ≤s 𝑥)
60	57, 59	sylibr 237	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → ∃𝑧 ∈ {𝑥}𝑥 ≤s 𝑧)
61	60	orcd 886	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → (∃𝑧 ∈ {𝑥}𝑥 ≤s 𝑧 ∨ ∃𝑤 ∈ ( R ‘𝑥)𝑤 ≤s ({𝑥} |s ∅)))
62		lltr 28033	. . . . . . . . . . . . . . . . 17 ⊢ ( L ‘𝑥) <<s ( R ‘𝑥)
63	62	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → ( L ‘𝑥) <<s ( R ‘𝑥))
64	31	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → {𝑥} <<s ∅)
65		lrcut 28075	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
66	51, 65	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
67	66	eqcomd 2769	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑥 = (( L ‘𝑥) |s ( R ‘𝑥)))
68		eqidd 2764	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → ({𝑥} |s ∅) = ({𝑥} |s ∅))
69	63, 64, 67, 68	ltsrecd 27973	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → (𝑥 <s ({𝑥} |s ∅) ↔ (∃𝑧 ∈ {𝑥}𝑥 ≤s 𝑧 ∨ ∃𝑤 ∈ ( R ‘𝑥)𝑤 ≤s ({𝑥} |s ∅))))
70	61, 69	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑥 <s ({𝑥} |s ∅))
71	50, 51, 52, 56, 70	leltstrd 27907	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 <s ({𝑥} |s ∅))
72		velsn 4606	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {({𝑥} |s ∅)} ↔ 𝑤 = ({𝑥} |s ∅))
73		breq2 5114	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ({𝑥} |s ∅) → (𝑧 <s 𝑤 ↔ 𝑧 <s ({𝑥} |s ∅)))
74	72, 73	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ {({𝑥} |s ∅)} → (𝑧 <s 𝑤 ↔ 𝑧 <s ({𝑥} |s ∅)))
75	71, 74	syl5ibrcom 250	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → (𝑤 ∈ {({𝑥} |s ∅)} → 𝑧 <s 𝑤))
76	75	3impia 1135	. . . . . . . . . . 11 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑤 ∈ {({𝑥} |s ∅)}) → 𝑧 <s 𝑤)
77	44, 46, 47, 49, 76	sltsd 27939	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 <<s {({𝑥} |s ∅)})
78		snelpwi 5427	. . . . . . . . . . 11 ⊢ (({𝑥} |s ∅) ∈ No → {({𝑥} |s ∅)} ∈ 𝒫 No )
79		nulsgts 27947	. . . . . . . . . . 11 ⊢ ({({𝑥} |s ∅)} ∈ 𝒫 No → {({𝑥} |s ∅)} <<s ∅)
80	48, 78, 79	3syl 19	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {({𝑥} |s ∅)} <<s ∅)
81	31, 41, 43, 77, 80	cofcut1d 28092	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) = (𝐴 |s ∅))
82	81	eqcomd 2769	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝐴 |s ∅) = ({𝑥} |s ∅))
83		simplrl 788	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 ⊆ ℕ0s)
84	83, 33	sseldd 3939	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ ℕ0s)
85	84	peano2n0sd 28502	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝑥 +s 1s ) ∈ ℕ0s)
86		n0cut 28505	. . . . . . . . . . 11 ⊢ ((𝑥 +s 1s ) ∈ ℕ0s → (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
87	85, 86	syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
88		1no 27981	. . . . . . . . . . . . 13 ⊢ 1s ∈ No
89		pncans 28243	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
90	28, 88, 89	sylancl 597	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
91	90	sneqd 4602	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {((𝑥 +s 1s ) -s 1s )} = {𝑥})
92	91	oveq1d 7427	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({((𝑥 +s 1s ) -s 1s )} |s ∅) = ({𝑥} |s ∅))
93	87, 92	eqtr2d 2799	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) = (𝑥 +s 1s ))
94	93, 85	eqeltrd 2863	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) ∈ ℕ0s)
95	82, 94	eqeltrd 2863	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝐴 |s ∅) ∈ ℕ0s)
96	95	expr 461	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥 → (𝐴 |s ∅) ∈ ℕ0s))
97	25, 96	sylbird 263	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (𝐴 |s ∅) ∈ ℕ0s))
98	97	rexlimdva 3166	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (𝐴 |s ∅) ∈ ℕ0s))
99	17, 98	mpd 16	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → (𝐴 |s ∅) ∈ ℕ0s)
100	99	ex 417	. 2 ⊢ (𝐴 ≠ ∅ → ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s))
101	6, 100	pm2.61ine 3041	1 ⊢ ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ⊆ wss 3906  ∅c0 4287  𝒫 cpw 4563  {csn 4590   class class class wbr 5110   Or wor 5570  ‘cfv 6538  (class class class)co 7412  Fincfn 8944   No csur 27782   <s clts 27783   ≤s cles 27886   <<s cslts 27928   |s ccuts 27930   0_s c0s 27976   1_s c1s 27977   L cleft 27996   R cright 27997   +_s cadds 28130   -_s csubs 28191  ℕ_0scn0s 28483

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-en 8945  df-fin 8948  df-no 27785  df-lts 27786  df-bday 27787  df-les 27887  df-slts 27929  df-cuts 27931  df-0s 27978  df-1s 27979  df-made 27998  df-old 27999  df-left 28001  df-right 28002  df-norec 28109  df-norec2 28120  df-adds 28131  df-negs 28192  df-subs 28193  df-n0s 28485

This theorem is referenced by:  onsfi  28527