Theorem n0sfincut 28298

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
n0sfincut	⊢ ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s)

Proof of Theorem n0sfincut

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

Step	Hyp	Ref	Expression
1		oveq1 7412	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 |s ∅) = (∅ |s ∅))
2		df-0s 27788	. . . . 5 ⊢ 0s = (∅ |s ∅)
3		0n0s 28274	. . . . 5 ⊢ 0s ∈ ℕ0s
4	2, 3	eqeltrri 2831	. . . 4 ⊢ (∅ |s ∅) ∈ ℕ0s
5	1, 4	eqeltrdi 2842	. . 3 ⊢ (𝐴 = ∅ → (𝐴 |s ∅) ∈ ℕ0s)
6	5	a1d 25	. 2 ⊢ (𝐴 = ∅ → ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s))
7		n0ssno 28265	. . . . . . . 8 ⊢ ℕ0s ⊆ No
8		sstr 3967	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℕ0s ∧ ℕ0s ⊆ No ) → 𝐴 ⊆ No )
9	7, 8	mpan2 691	. . . . . . 7 ⊢ (𝐴 ⊆ ℕ0s → 𝐴 ⊆ No )
10		sltso 27640	. . . . . . 7 ⊢ <s Or No
11		soss 5581	. . . . . . 7 ⊢ (𝐴 ⊆ No → ( <s Or No → <s Or 𝐴))
12	9, 10, 11	mpisyl 21	. . . . . 6 ⊢ (𝐴 ⊆ ℕ0s → <s Or 𝐴)
13	12	ad2antrl 728	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → <s Or 𝐴)
14		simprr 772	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ Fin)
15		simpl 482	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → 𝐴 ≠ ∅)
16		fimax2g 9294	. . . . 5 ⊢ (( <s Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
17	13, 14, 15, 16	syl3anc 1373	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
18	9	ad2antrl 728	. . . . . . . . . 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 3958	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ No )
21	18	sselda 3958	. . . . . . . . 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		slenlt 27716	. . . . . . . 8 ⊢ ((𝑦 ∈ No ∧ 𝑥 ∈ No ) → (𝑦 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑦))
24	20, 22, 23	syl2anc 584	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑦))
25	24	ralbidva 3161	. . . . . 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 3953	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No )
28	18, 26, 27	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ No )
29		snelpwi 5418	. . . . . . . . . . 11 ⊢ (𝑥 ∈ No → {𝑥} ∈ 𝒫 No )
30		nulssgt 27762	. . . . . . . . . . 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 5123	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑥 ≤s 𝑤 ↔ 𝑥 ≤s 𝑥))
33		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ 𝐴)
34		slerflex 27727	. . . . . . . . . . . . 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 3604	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ∃𝑤 ∈ 𝐴 𝑥 ≤s 𝑤)
37		vex 3463	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
38		breq1 5122	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ≤s 𝑤 ↔ 𝑥 ≤s 𝑤))
39	38	rexbidv 3164	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (∃𝑤 ∈ 𝐴 𝑧 ≤s 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑥 ≤s 𝑤))
40	37, 39	ralsn 4657	. . . . . . . . . . 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 4488	. . . . . . . . . . 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 777	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 ∈ Fin)
45		snex 5406	. . . . . . . . . . . 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	scutcld 27767	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) ∈ No )
49	48	snssd 4785	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {({𝑥} |s ∅)} ⊆ No )
50	47	sselda 3958	. . . . . . . . . . . . . 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 5122	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝑦 ≤s 𝑥 ↔ 𝑧 ≤s 𝑥))
54		simplrr 777	. . . . . . . . . . . . . . 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 3602	. . . . . . . . . . . . . 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 5123	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑥 → (𝑥 ≤s 𝑧 ↔ 𝑥 ≤s 𝑥))
59	37, 58	rexsn 4658	. . . . . . . . . . . . . . . . 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 873	. . . . . . . . . . . . . . 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		lltropt 27836	. . . . . . . . . . . . . . . . 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 27867	. . . . . . . . . . . . . . . . . 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 2741	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑥 = (( L ‘𝑥) |s ( R ‘𝑥)))
68		eqidd 2736	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → ({𝑥} |s ∅) = ({𝑥} |s ∅))
69		sltrec 27784	. . . . . . . . . . . . . . . 16 ⊢ (((( L ‘𝑥) <<s ( R ‘𝑥) ∧ {𝑥} <<s ∅) ∧ (𝑥 = (( L ‘𝑥) |s ( R ‘𝑥)) ∧ ({𝑥} |s ∅) = ({𝑥} |s ∅))) → (𝑥 <s ({𝑥} |s ∅) ↔ (∃𝑧 ∈ {𝑥}𝑥 ≤s 𝑧 ∨ ∃𝑤 ∈ ( R ‘𝑥)𝑤 ≤s ({𝑥} |s ∅))))
70	63, 64, 67, 68, 69	syl22anc 838	. . . . . . . . . . . . . . 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 ∅))))
71	61, 70	mpbird 257	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑥 <s ({𝑥} |s ∅))
72	50, 51, 52, 56, 71	slelttrd 27725	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 <s ({𝑥} |s ∅))
73		velsn 4617	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {({𝑥} |s ∅)} ↔ 𝑤 = ({𝑥} |s ∅))
74		breq2 5123	. . . . . . . . . . . . . 14 ⊢ (𝑤 = ({𝑥} |s ∅) → (𝑧 <s 𝑤 ↔ 𝑧 <s ({𝑥} |s ∅)))
75	73, 74	sylbi 217	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ {({𝑥} |s ∅)} → (𝑧 <s 𝑤 ↔ 𝑧 <s ({𝑥} |s ∅)))
76	72, 75	syl5ibrcom 247	. . . . . . . . . . . 12 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → (𝑤 ∈ {({𝑥} |s ∅)} → 𝑧 <s 𝑤))
77	76	3impia 1117	. . . . . . . . . . 11 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴 ∧ 𝑤 ∈ {({𝑥} |s ∅)}) → 𝑧 <s 𝑤)
78	44, 46, 47, 49, 77	ssltd 27755	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 <<s {({𝑥} |s ∅)})
79		snelpwi 5418	. . . . . . . . . . 11 ⊢ (({𝑥} |s ∅) ∈ No → {({𝑥} |s ∅)} ∈ 𝒫 No )
80		nulssgt 27762	. . . . . . . . . . 11 ⊢ ({({𝑥} |s ∅)} ∈ 𝒫 No → {({𝑥} |s ∅)} <<s ∅)
81	48, 79, 80	3syl 18	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {({𝑥} |s ∅)} <<s ∅)
82	31, 41, 43, 78, 81	cofcut1d 27881	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) = (𝐴 |s ∅))
83	82	eqcomd 2741	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝐴 |s ∅) = ({𝑥} |s ∅))
84		simplrl 776	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 ⊆ ℕ0s)
85	84, 33	sseldd 3959	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ ℕ0s)
86		peano2n0s 28275	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ0s → (𝑥 +s 1s ) ∈ ℕ0s)
87	85, 86	syl 17	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝑥 +s 1s ) ∈ ℕ0s)
88		n0scut 28278	. . . . . . . . . . 11 ⊢ ((𝑥 +s 1s ) ∈ ℕ0s → (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
89	87, 88	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝑥 +s 1s ) = ({((𝑥 +s 1s ) -s 1s )} |s ∅))
90		1sno 27791	. . . . . . . . . . . . 13 ⊢ 1s ∈ No
91		pncans 28028	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No ) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
92	28, 90, 91	sylancl 586	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ((𝑥 +s 1s ) -s 1s ) = 𝑥)
93	92	sneqd 4613	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {((𝑥 +s 1s ) -s 1s )} = {𝑥})
94	93	oveq1d 7420	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({((𝑥 +s 1s ) -s 1s )} |s ∅) = ({𝑥} |s ∅))
95	89, 94	eqtr2d 2771	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) = (𝑥 +s 1s ))
96	95, 87	eqeltrd 2834	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) ∈ ℕ0s)
97	83, 96	eqeltrd 2834	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝐴 |s ∅) ∈ ℕ0s)
98	97	expr 456	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥 → (𝐴 |s ∅) ∈ ℕ0s))
99	25, 98	sylbird 260	. . . . 5 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (𝐴 |s ∅) ∈ ℕ0s))
100	99	rexlimdva 3141	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦 → (𝐴 |s ∅) ∈ ℕ0s))
101	17, 100	mpd 15	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → (𝐴 |s ∅) ∈ ℕ0s)
102	101	ex 412	. 2 ⊢ (𝐴 ≠ ∅ → ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s))
103	6, 102	pm2.61ine 3015	1 ⊢ ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  Vcvv 3459   ⊆ wss 3926  ∅c0 4308  𝒫 cpw 4575  {csn 4601   class class class wbr 5119   Or wor 5560  ‘cfv 6531  (class class class)co 7405  Fincfn 8959   No csur 27603   <s cslt 27604   ≤s csle 27708   <<s csslt 27744   |s cscut 27746   0_s c0s 27786   1_s c1s 27787   L cleft 27805   R cright 27806   +_s cadds 27918   -_s csubs 27978  ℕ_0scnn0s 28258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-ot 4610  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-nadd 8678  df-en 8960  df-fin 8963  df-no 27606  df-slt 27607  df-bday 27608  df-sle 27709  df-sslt 27745  df-scut 27747  df-0s 27788  df-1s 27789  df-made 27807  df-old 27808  df-left 27810  df-right 27811  df-norec 27897  df-norec2 27908  df-adds 27919  df-negs 27979  df-subs 27980  df-n0s 28260

This theorem is referenced by:  onsfi  28299