Theorem n0fincut 28746

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 7427	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 |s ∅) = (∅ |s ∅))
2		df-0s 28198	. . . . 5 ⊢ 0s = (∅ |s ∅)
3		0n0s 28720	. . . . 5 ⊢ 0s ∈ ℕ0s
4	2, 3	eqeltrri 2857	. . . 4 ⊢ (∅ |s ∅) ∈ ℕ0s
5	1, 4	eqeltrdi 2868	. . 3 ⊢ (𝐴 = ∅ → (𝐴 |s ∅) ∈ ℕ0s)
6	5	a1d 26	. 2 ⊢ (𝐴 = ∅ → ((𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin) → (𝐴 |s ∅) ∈ ℕ0s))
7		n0ssno 28711	. . . . . . . 8 ⊢ ℕ0s ⊆ No
8		sstr 3916	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℕ0s ∧ ℕ0s ⊆ No) → 𝐴 ⊆ No)
9	7, 8	mpan2 704	. . . . . . 7 ⊢ (𝐴 ⊆ ℕ0s → 𝐴 ⊆ No)
10		ltsso 28038	. . . . . . 7 ⊢ <s Or No
11		soss 5578	. . . . . . 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 9280	. . . . 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 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 3908	. . . . . . . 8 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ No)
21	18	sselda 3908	. . . . . . . . 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 28114	. . . . . . . 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 3183	. . . . . 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 3903	. . . . . . . . . . . 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 5411	. . . . . . . . . . 11 ⊢ (𝑥 ∈ No → {𝑥} ∈ 𝒫 No)
30		nulsgts 28167	. . . . . . . . . . 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 5106	. . . . . . . . . . . 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 28129	. . . . . . . . . . . . 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 3579	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ∃𝑤 ∈ 𝐴 𝑥 ≤s 𝑤)
37		vex 3454	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
38		breq1 5105	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 ≤s 𝑤 ↔ 𝑥 ≤s 𝑤))
39	38	rexbidv 3186	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → (∃𝑤 ∈ 𝐴 𝑧 ≤s 𝑤 ↔ ∃𝑤 ∈ 𝐴 𝑥 ≤s 𝑤))
40	37, 39	ralsn 4638	. . . . . . . . . . 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 4453	. . . . . . . . . . 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 5396	. . . . . . . . . . . 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 28174	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) ∈ No)
49	48	snssd 4743	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {({𝑥} |s ∅)} ⊆ No)
50	47	sselda 3908	. . . . . . . . . . . . . 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 5105	. . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . 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 5106	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = 𝑥 → (𝑥 ≤s 𝑧 ↔ 𝑥 ≤s 𝑥))
59	37, 58	rexsn 4639	. . . . . . . . . . . . . . . . 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 28253	. . . . . . . . . . . . . . . . 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 28295	. . . . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑥 = ((L‘𝑥) |s (R‘𝑥)))
68		eqidd 2761	. . . . . . . . . . . . . . . 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 28193	. . . . . . . . . . . . . . 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 28127	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) ∧ 𝑧 ∈ 𝐴) → 𝑧 <s ({𝑥} |s ∅))
72		velsn 4596	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {({𝑥} |s ∅)} ↔ 𝑤 = ({𝑥} |s ∅))
73		breq2 5106	. . . . . . . . . . . . . 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 28159	. . . . . . . . . 10 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝐴 <<s {({𝑥} |s ∅)})
78		snelpwi 5411	. . . . . . . . . . 11 ⊢ (({𝑥} |s ∅) ∈ No → {({𝑥} |s ∅)} ∈ 𝒫 No)
79		nulsgts 28167	. . . . . . . . . . 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 28312	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) = (𝐴 |s ∅))
82	81	eqcomd 2766	. . . . . . . 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 3909	. . . . . . . . . . . 12 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → 𝑥 ∈ ℕ0s)
85	84	peano2n0sd 28722	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → (𝑥 +s 1s ) ∈ ℕ0s)
86		n0cut 28725	. . . . . . . . . . 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 28201	. . . . . . . . . . . . 13 ⊢ 1s ∈ No
89		pncans 28463	. . . . . . . . . . . . 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 4592	. . . . . . . . . . 11 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → {((𝑥 +s 1s ) −s 1s )} = {𝑥})
92	91	oveq1d 7435	. . . . . . . . . 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 2796	. . . . . . . . 9 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) = (𝑥 +s 1s ))
94	93, 85	eqeltrd 2860	. . . . . . . 8 ⊢ (((𝐴 ≠ ∅ ∧ (𝐴 ⊆ ℕ0s ∧ 𝐴 ∈ Fin)) ∧ (𝑥 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 𝑦 ≤s 𝑥)) → ({𝑥} |s ∅) ∈ ℕ0s)
95	82, 94	eqeltrd 2860	. . . . . . 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 3163	. . . 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 3038	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 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  𝒫 cpw 4556  {csn 4584   class class class wbr 5102   Or wor 5557  ‘cfv 6538  (class class class)co 7420  Fincfn 8976  Nocsur 28002   <s clts 28003   ≤s cles 28106   <<s cslts 28148   |s ccuts 28150   0_s c0s 28196   1_s c1s 28197  Lcleft 28216  Rcright 28217   +_s cadds 28350   −_s csubs 28411  ℕ_0scn0s 28703

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-ot 4858  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  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 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-rdg 8421  df-1o 8479  df-2o 8480  df-nadd 8678  df-en 8977  df-fin 8980  df-no 28005  df-lts 28006  df-bday 28007  df-les 28107  df-slts 28149  df-cuts 28151  df-0s 28198  df-1s 28199  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-norec2 28340  df-adds 28351  df-negs 28412  df-subs 28413  df-n0s 28705

This theorem is used by:  onsfi  28747