Theorem n0fincut 28347

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 7374	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 |s ∅) = (∅ |s ∅))
2		df-0s 27799	. . . . 5 ⊢ 0s = (∅ |s ∅)
3		0n0s 28321	. . . . 5 ⊢ 0s ∈ ℕ0s
4	2, 3	eqeltrri 2833	. . . 4 ⊢ (∅ |s ∅) ∈ ℕ0s
5	1, 4	eqeltrdi 2844	. . 3 ⊢ (𝐴 = ∅ → (𝐴 |s ∅) ∈ ℕ0s)
6	5	a1d 25	. 2 ⊢ (𝐴 = ∅ → ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s))
7		n0ssno 28312	. . . . . . . 8 ⊢ ℕ0s ⊆ No
8		sstr 3930	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℕ0s ∧ ℕ0s ⊆ No ) → 𝐴 ⊆ No )
9	7, 8	mpan2 692	. . . . . . 7 ⊢ (𝐴 ⊆ ℕ0s → 𝐴 ⊆ No )
10		ltsso 27640	. . . . . . 7 ⊢ <s Or No
11		soss 5559	. . . . . . 7 ⊢ (𝐴 ⊆ No → ( <s Or No → <s Or 𝐴))
12	9, 10, 11	mpisyl 21	. . . . . 6 ⊢ (𝐴 ⊆ ℕ0s → <s Or 𝐴)
13	12	ad2antrl 729	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → <s Or 𝐴)
14		simprr 773	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ Fin)
15		simpl 482	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → 𝐴 ≠ ∅)
16		fimax2g 9196	. . . . 5 ⊢ (( <s Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
17	13, 14, 15, 16	syl3anc 1374	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
18	9	ad2antrl 729	. . . . . . . . . 10 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → 𝐴 ⊆ No )
19	18	adantr 480	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ No )
20	19	sselda 3921	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ No )
21	18	sselda 3921	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No )
22	21	adantr 480	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ No )
23		lenlts 27716	. . . . . . . 8 ⊢ ((𝑦 ∈ No ∧ 𝑥 ∈ No ) → (𝑦 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑦))
24	20, 22, 23	syl2anc 585	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑦))
25	24	ralbidva 3158	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
26		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥) → 𝑥 ∈ 𝐴)
27		ssel2 3916	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No )
28	18, 26, 27	syl2an 597	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ No )
29		snelpwi 5396	. . . . . . . . . . 11 ⊢ (𝑥 ∈ No → {𝑥} ∈ 𝒫 No )
30		nulsgts 27768	. . . . . . . . . . 11 ⊢ ({𝑥} ∈ 𝒫 No → {𝑥} <<s ∅)
31	28, 29, 30	3syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {𝑥} <<s ∅)
32		breq2 5089	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑥 ≤s 𝑤 ↔ 𝑥 ≤s 𝑥))
33		simprl 771	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ 𝐴)
34		lesid 27731	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ No → 𝑥 ≤s 𝑥)
35	28, 34	syl 17	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ≤s 𝑥)
36	32, 33, 35	rspcedvdw 3567	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ∃𝑤 ∈ 𝐴 𝑥 ≤s 𝑤)
37		vex 3433	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
38		breq1 5088	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ≤s 𝑤 ↔ 𝑥 ≤s 𝑤))
39	38	rexbidv 3161	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (∃𝑤 ∈ 𝐴 𝑧 ≤s 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑥 ≤s 𝑤))
40	37, 39	ralsn 4625	. . . . . . . . . . 11 ⊢ (∀𝑧 ∈ {𝑥}∃𝑤 ∈ 𝐴 𝑧 ≤s 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑥 ≤s 𝑤)
41	36, 40	sylibr 234	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ∀𝑧 ∈ {𝑥}∃𝑤 ∈ 𝐴 𝑧 ≤s 𝑤)
42		ral0 4438	. . . . . . . . . . 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 778	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 ∈ Fin)
45		snex 5381	. . . . . . . . . . . 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 480	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 ⊆ No )
48	31	cutscld 27775	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) ∈ No )
49	48	snssd 4730	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {({𝑥} |s ∅)} ⊆ No )
50	47	sselda 3921	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ No )
51	28	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ No )
52	48	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → ({𝑥} |s ∅) ∈ No )
53		breq1 5088	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝑦 ≤s 𝑥 ↔ 𝑧 ≤s 𝑥))
54		simplrr 778	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)
55		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
56	53, 54, 55	rspcdva 3565	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ≤s 𝑥)
57	51, 34	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑥 ≤s 𝑥)
58		breq2 5089	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑥 → (𝑥 ≤s 𝑧 ↔ 𝑥 ≤s 𝑥))
59	37, 58	rexsn 4626	. . . . . . . . . . . . . . . . 17 ⊢ (∃𝑧 ∈ {𝑥}𝑥 ≤s 𝑧 ↔ 𝑥 ≤s 𝑥)
60	57, 59	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → ∃𝑧 ∈ {𝑥}𝑥 ≤s 𝑧)
61	60	orcd 874	. . . . . . . . . . . . . . 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 27854	. . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → {𝑥} <<s ∅)
65		lrcut 27896	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
66	51, 65	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
67	66	eqcomd 2742	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑥 = (( L ‘𝑥) |s ( R ‘𝑥)))
68		eqidd 2737	. . . . . . . . . . . . . . . 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 27794	. . . . . . . . . . . . . . 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 257	. . . . . . . . . . . . . 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 27729	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 <s ({𝑥} |s ∅))
72		velsn 4583	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {({𝑥} |s ∅)} ↔ 𝑤 = ({𝑥} |s ∅))
73		breq2 5089	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ({𝑥} |s ∅) → (𝑧 <s 𝑤 ↔ 𝑧 <s ({𝑥} |s ∅)))
74	72, 73	sylbi 217	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ {({𝑥} |s ∅)} → (𝑧 <s 𝑤 ↔ 𝑧 <s ({𝑥} |s ∅)))
75	71, 74	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → (𝑤 ∈ {({𝑥} |s ∅)} → 𝑧 <s 𝑤))
76	75	3impia 1118	. . . . . . . . . . 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 27760	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 <<s {({𝑥} |s ∅)})
78		snelpwi 5396	. . . . . . . . . . 11 ⊢ (({𝑥} |s ∅) ∈ No → {({𝑥} |s ∅)} ∈ 𝒫 No )
79		nulsgts 27768	. . . . . . . . . . 11 ⊢ ({({𝑥} |s ∅)} ∈ 𝒫 No → {({𝑥} |s ∅)} <<s ∅)
80	48, 78, 79	3syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {({𝑥} |s ∅)} <<s ∅)
81	31, 41, 43, 77, 80	cofcut1d 27913	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) = (𝐴 |s ∅))
82	81	eqcomd 2742	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝐴 |s ∅) = ({𝑥} |s ∅))
83		simplrl 777	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 ⊆ ℕ0s)
84	83, 33	sseldd 3922	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ ℕ0s)
85	84	peano2n0sd 28323	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝑥 +s 1s ) ∈ ℕ0s)
86		n0cut 28326	. . . . . . . . . . 11 ⊢ ((𝑥 +s 1s ) ∈ ℕ0s → (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
87	85, 86	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
88		1no 27802	. . . . . . . . . . . . 13 ⊢ 1s ∈ No
89		pncans 28064	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
90	28, 88, 89	sylancl 587	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
91	90	sneqd 4579	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {((𝑥 +s 1s ) -s 1s )} = {𝑥})
92	91	oveq1d 7382	. . . . . . . . . 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 2772	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) = (𝑥 +s 1s ))
94	93, 85	eqeltrd 2836	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) ∈ ℕ0s)
95	82, 94	eqeltrd 2836	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝐴 |s ∅) ∈ ℕ0s)
96	95	expr 456	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥 → (𝐴 |s ∅) ∈ ℕ0s))
97	25, 96	sylbird 260	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (𝐴 |s ∅) ∈ ℕ0s))
98	97	rexlimdva 3138	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (𝐴 |s ∅) ∈ ℕ0s))
99	17, 98	mpd 15	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → (𝐴 |s ∅) ∈ ℕ0s)
100	99	ex 412	. 2 ⊢ (𝐴 ≠ ∅ → ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s))
101	6, 100	pm2.61ine 3015	1 ⊢ ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  Vcvv 3429   ⊆ wss 3889  ∅c0 4273  𝒫 cpw 4541  {csn 4567   class class class wbr 5085   Or wor 5538  ‘cfv 6498  (class class class)co 7367  Fincfn 8893   No csur 27603   <s clts 27604   ≤s cles 27708   <<s cslts 27749   |s ccuts 27751   0_s c0s 27797   1_s c1s 27798   L cleft 27817   R cright 27818   +_s cadds 27951   -_s csubs 28012  ℕ_0scn0s 28304

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-nadd 8602  df-en 8894  df-fin 8897  df-no 27606  df-lts 27607  df-bday 27608  df-les 27709  df-slts 27750  df-cuts 27752  df-0s 27799  df-1s 27800  df-made 27819  df-old 27820  df-left 27822  df-right 27823  df-norec 27930  df-norec2 27941  df-adds 27952  df-negs 28013  df-subs 28014  df-n0s 28306

This theorem is referenced by:  onsfi  28348