Theorem n0s0m1 28755

Description: Every non-negative surreal integer is either zero or a successor. (Contributed by Scott Fenton, 26-May-2025.)

Assertion

Ref	Expression
n0s0m1	⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 −s 1s ) ∈ ℕ0s))

Proof of Theorem n0s0m1

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

Step	Hyp	Ref	Expression
1		eqeq1 2764	. . 3 ⊢ (𝑥 = 0s → (𝑥 = 0s ↔ 0s = 0s ))
2		oveq1 7429	. . . 4 ⊢ (𝑥 = 0s → (𝑥 −s 1s ) = ( 0s −s 1s ))
3	2	eleq1d 2845	. . 3 ⊢ (𝑥 = 0s → ((𝑥 −s 1s ) ∈ ℕ0s ↔ ( 0s −s 1s ) ∈ ℕ0s))
4	1, 3	orbi12d 932	. 2 ⊢ (𝑥 = 0s → ((𝑥 = 0s ∨ (𝑥 −s 1s ) ∈ ℕ0s) ↔ ( 0s = 0s ∨ ( 0s −s 1s ) ∈ ℕ0s)))
5		eqeq1 2764	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 0s ↔ 𝑦 = 0s ))
6		oveq1 7429	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 −s 1s ) = (𝑦 −s 1s ))
7	6	eleq1d 2845	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 −s 1s ) ∈ ℕ0s ↔ (𝑦 −s 1s ) ∈ ℕ0s))
8	5, 7	orbi12d 932	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 0s ∨ (𝑥 −s 1s ) ∈ ℕ0s) ↔ (𝑦 = 0s ∨ (𝑦 −s 1s ) ∈ ℕ0s)))
9		eqeq1 2764	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 = 0s ↔ (𝑦 +s 1s ) = 0s ))
10		oveq1 7429	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 −s 1s ) = ((𝑦 +s 1s ) −s 1s ))
11	10	eleq1d 2845	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 −s 1s ) ∈ ℕ0s ↔ ((𝑦 +s 1s ) −s 1s ) ∈ ℕ0s))
12	9, 11	orbi12d 932	. 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 = 0s ∨ (𝑥 −s 1s ) ∈ ℕ0s) ↔ ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) −s 1s ) ∈ ℕ0s)))
13		eqeq1 2764	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 0s ↔ 𝐴 = 0s ))
14		oveq1 7429	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 −s 1s ) = (𝐴 −s 1s ))
15	14	eleq1d 2845	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 −s 1s ) ∈ ℕ0s ↔ (𝐴 −s 1s ) ∈ ℕ0s))
16	13, 15	orbi12d 932	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 0s ∨ (𝑥 −s 1s ) ∈ ℕ0s) ↔ (𝐴 = 0s ∨ (𝐴 −s 1s ) ∈ ℕ0s)))
17		eqid 2760	. . 3 ⊢ 0s = 0s
18	17	orci 879	. 2 ⊢ ( 0s = 0s ∨ ( 0s −s 1s ) ∈ ℕ0s)
19		n0no 28716	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No)
20		1no 28203	. . . . . 6 ⊢ 1s ∈ No
21		pncans 28465	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ 1s ∈ No) → ((𝑦 +s 1s ) −s 1s ) = 𝑦)
22	19, 20, 21	sylancl 598	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) −s 1s ) = 𝑦)
23		id 23	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℕ0s)
24	22, 23	eqeltrd 2860	. . . 4 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) −s 1s ) ∈ ℕ0s)
25	24	olcd 888	. . 3 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) −s 1s ) ∈ ℕ0s))
26	25	a1d 26	. 2 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 = 0s ∨ (𝑦 −s 1s ) ∈ ℕ0s) → ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) −s 1s ) ∈ ℕ0s)))
27	4, 8, 12, 16, 18, 26	n0sind 28726	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 −s 1s ) ∈ ℕ0s))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1569   ∈ wcel 2144  (class class class)co 7422  Nocsur 28004   0_s c0s 28198   1_s c1s 28199   +_s cadds 28352   −_s csubs 28413  ℕ_0scn0s 28705

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 7753

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 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  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 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-2o 8482  df-nadd 8680  df-no 28007  df-lts 28008  df-bday 28009  df-les 28109  df-slts 28151  df-cuts 28153  df-0s 28200  df-1s 28201  df-made 28220  df-old 28221  df-left 28223  df-right 28224  df-norec 28331  df-norec2 28342  df-adds 28353  df-negs 28414  df-subs 28415  df-n0s 28707

This theorem is used by:  n0subs  28756