Theorem n0fincut 28774

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 7453	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 |s ∅) = (∅ |s ∅))
2		df-0s 28226	. . . . 5 ⊢ 0s = (∅ |s ∅)
3		0n0s 28748	. . . . 5 ⊢ 0s ∈ ℕ0s
4	2, 3	eqeltrri 2861	. . . 4 ⊢ (∅ |s ∅) ∈ ℕ0s
5	1, 4	eqeltrdi 2872	. . 3 ⊢ (𝐴 = ∅ → (𝐴 |s ∅) ∈ ℕ0s)
6	5	a1d 26	. 2 ⊢ (𝐴 = ∅ → ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s))
7		n0ssno 28739	. . . . . . . 8 ⊢ ℕ0s ⊆ No
8		sstr 3921	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℕ0s ∧ ℕ0s ⊆ No) → 𝐴 ⊆ No)
9	7, 8	mpan2 704	. . . . . . 7 ⊢ (𝐴 ⊆ ℕ0s → 𝐴 ⊆ No)
10		ltsso 28066	. . . . . . 7 ⊢ <s Or No
11		soss 5584	. . . . . . 7 ⊢ (𝐴 ⊆ No → ( <s Or No → <s Or 𝐴))
12	9, 10, 11	mpisyl 22	. . . . . 6 ⊢ (𝐴 ⊆ ℕ0s → <s Or 𝐴)
13	12	ad2antrl 741	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → <s Or 𝐴)
14		simprr 785	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → 𝐴 ∈ Fin)
15		simpl 488	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → 𝐴 ≠ ∅)
16		fimax2g 9308	. . . . 5 ⊢ (( <s Or 𝐴 ∧ 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
17	13, 14, 15, 16	syl3anc 1399	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦)
18	9	ad2antrl 741	. . . . . . . . . 10 ⊢ ((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) → 𝐴 ⊆ No)
19	18	adantr 486	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → 𝐴 ⊆ No)
20	19	sselda 3913	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ No)
21	18	sselda 3913	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No)
22	21	adantr 486	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ No)
23		lenlts 28142	. . . . . . . 8 ⊢ ((𝑦 ∈ No ∧ 𝑥 ∈ No) → (𝑦 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑦))
24	20, 22, 23	syl2anc 596	. . . . . . 7 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 ≤s 𝑥 ↔ ¬ 𝑥 <s 𝑦))
25	24	ralbidva 3187	. . . . . 6 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦))
26		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥) → 𝑥 ∈ 𝐴)
27		ssel2 3908	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ No ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ No)
28	18, 26, 27	syl2an 608	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ No)
29		snelpwi 5417	. . . . . . . . . . 11 ⊢ (𝑥 ∈ No → {𝑥} ∈ 𝒫 No)
30		nulsgts 28195	. . . . . . . . . . 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 5110	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝑥 ≤s 𝑤 ↔ 𝑥 ≤s 𝑥))
33		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ 𝐴)
34		lesid 28157	. . . . . . . . . . . . 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 3583	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ∃𝑤 ∈ 𝐴 𝑥 ≤s 𝑤)
37		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
38		breq1 5109	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ≤s 𝑤 ↔ 𝑥 ≤s 𝑤))
39	38	rexbidv 3190	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (∃𝑤 ∈ 𝐴 𝑧 ≤s 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑥 ≤s 𝑤))
40	37, 39	ralsn 4642	. . . . . . . . . . 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 4457	. . . . . . . . . . 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 790	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 ∈ Fin)
45		snex 5402	. . . . . . . . . . . 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 486	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 ⊆ No)
48	31	cutscld 28202	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) ∈ No)
49	48	snssd 4747	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {({𝑥} |s ∅)} ⊆ No)
50	47	sselda 3913	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ No)
51	28	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑥 ∈ No)
52	48	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → ({𝑥} |s ∅) ∈ No)
53		breq1 5109	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑧 → (𝑦 ≤s 𝑥 ↔ 𝑧 ≤s 𝑥))
54		simplrr 790	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)
55		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 ∈ 𝐴)
56	53, 54, 55	rspcdva 3581	. . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑥 → (𝑥 ≤s 𝑧 ↔ 𝑥 ≤s 𝑥))
59	37, 58	rexsn 4643	. . . . . . . . . . . . . . . . 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 887	. . . . . . . . . . . . . . 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 28281	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → {𝑥} <<s ∅)
65		lrcut 28323	. . . . . . . . . . . . . . . . . 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 28221	. . . . . . . . . . . . . . 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 28155	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 <s ({𝑥} |s ∅))
72		velsn 4600	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {({𝑥} |s ∅)} ↔ 𝑤 = ({𝑥} |s ∅))
73		breq2 5110	. . . . . . . . . . . . . 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 28187	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 <<s {({𝑥} |s ∅)})
78		snelpwi 5417	. . . . . . . . . . 11 ⊢ (({𝑥} |s ∅) ∈ No → {({𝑥} |s ∅)} ∈ 𝒫 No)
79		nulsgts 28195	. . . . . . . . . . 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 28340	. . . . . . . . 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 789	. . . . . . . . . . . . 13 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 ⊆ ℕ0s)
84	83, 33	sseldd 3914	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ ℕ0s)
85	84	peano2n0sd 28750	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝑥 +s 1s ) ∈ ℕ0s)
86		n0cut 28753	. . . . . . . . . . 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 28229	. . . . . . . . . . . . 13 ⊢ 1s ∈ No
89		pncans 28491	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ No ∧ 1s ∈ No) → ((𝑥 +s 1s ) −s 1s ) = 𝑥)
90	28, 88, 89	sylancl 598	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ((𝑥 +s 1s ) −s 1s ) = 𝑥)
91	90	sneqd 4596	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {((𝑥 +s 1s ) −s 1s )} = {𝑥})
92	91	oveq1d 7461	. . . . . . . . . 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 2864	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) ∈ ℕ0s)
95	82, 94	eqeltrd 2864	. . . . . . 7 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝐴 |s ∅) ∈ ℕ0s)
96	95	expr 462	. . . . . 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 3167	. . . 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 418	. 2 ⊢ (𝐴 ≠ ∅ → ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s))
101	6, 100	pm2.61ine 3042	1 ⊢ ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900  ∅c0 4281  𝒫 cpw 4560  {csn 4588   class class class wbr 5106   Or wor 5563  ‘cfv 6561  (class class class)co 7446  Fincfn 9004  Nocsur 28030   <s clts 28031   ≤s cles 28134   <<s cslts 28176   |s ccuts 28178   0_s c0s 28224   1_s c1s 28225  Lcleft 28244  Rcright 28245   +_s cadds 28378   −_s csubs 28439  ℕ_0scn0s 28731

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-nadd 8706  df-en 9005  df-fin 9008  df-no 28033  df-lts 28034  df-bday 28035  df-les 28135  df-slts 28177  df-cuts 28179  df-0s 28226  df-1s 28227  df-made 28246  df-old 28247  df-left 28249  df-right 28250  df-norec 28357  df-norec2 28368  df-adds 28379  df-negs 28440  df-subs 28441  df-n0s 28733

This theorem is used by:  onsfi  28775