Theorem nn1m1nns 28637

Description: Every positive surreal integer is either one or a successor. (Contributed by Scott Fenton, 8-Nov-2025.)

Assertion

Ref	Expression
nn1m1nns	⊢ (𝐴 ∈ ℕs → (𝐴 = 1s ∨ (𝐴 -s 1s ) ∈ ℕs))

Proof of Theorem nn1m1nns

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

Step	Hyp	Ref	Expression
1		eqeq1 2766	. . 3 ⊢ (𝑥 = 1s → (𝑥 = 1s ↔ 1s = 1s ))
2		oveq1 7423	. . . 4 ⊢ (𝑥 = 1s → (𝑥 -s 1s ) = ( 1s -s 1s ))
3	2	eleq1d 2847	. . 3 ⊢ (𝑥 = 1s → ((𝑥 -s 1s ) ∈ ℕs ↔ ( 1s -s 1s ) ∈ ℕs))
4	1, 3	orbi12d 932	. 2 ⊢ (𝑥 = 1s → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ ( 1s = 1s ∨ ( 1s -s 1s ) ∈ ℕs)))
5		eqeq1 2766	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1s ↔ 𝑦 = 1s ))
6		oveq1 7423	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 -s 1s ) = (𝑦 -s 1s ))
7	6	eleq1d 2847	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 -s 1s ) ∈ ℕs ↔ (𝑦 -s 1s ) ∈ ℕs))
8	5, 7	orbi12d 932	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ (𝑦 = 1s ∨ (𝑦 -s 1s ) ∈ ℕs)))
9		eqeq1 2766	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 = 1s ↔ (𝑦 +s 1s ) = 1s ))
10		oveq1 7423	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 1s ) = ((𝑦 +s 1s ) -s 1s ))
11	10	eleq1d 2847	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 -s 1s ) ∈ ℕs ↔ ((𝑦 +s 1s ) -s 1s ) ∈ ℕs))
12	9, 11	orbi12d 932	. 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ ((𝑦 +s 1s ) = 1s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕs)))
13		eqeq1 2766	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1s ↔ 𝐴 = 1s ))
14		oveq1 7423	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 -s 1s ) = (𝐴 -s 1s ))
15	14	eleq1d 2847	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 -s 1s ) ∈ ℕs ↔ (𝐴 -s 1s ) ∈ ℕs))
16	13, 15	orbi12d 932	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ (𝐴 = 1s ∨ (𝐴 -s 1s ) ∈ ℕs)))
17		eqid 2762	. . 3 ⊢ 1s = 1s
18	17	orci 879	. 2 ⊢ ( 1s = 1s ∨ ( 1s -s 1s ) ∈ ℕs)
19		nnno 28587	. . . . . 6 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No )
20		1no 28073	. . . . . 6 ⊢ 1s ∈ No
21		pncans 28335	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ 1s ∈ No ) → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
22	19, 20, 21	sylancl 598	. . . . 5 ⊢ (𝑦 ∈ ℕs → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
23		id 23	. . . . 5 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ ℕs)
24	22, 23	eqeltrd 2862	. . . 4 ⊢ (𝑦 ∈ ℕs → ((𝑦 +s 1s ) -s 1s ) ∈ ℕs)
25	24	olcd 888	. . 3 ⊢ (𝑦 ∈ ℕs → ((𝑦 +s 1s ) = 1s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕs))
26	25	a1d 26	. 2 ⊢ (𝑦 ∈ ℕs → ((𝑦 = 1s ∨ (𝑦 -s 1s ) ∈ ℕs) → ((𝑦 +s 1s ) = 1s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕs)))
27	4, 8, 12, 16, 18, 26	nnsind 28636	1 ⊢ (𝐴 ∈ ℕs → (𝐴 = 1s ∨ (𝐴 -s 1s ) ∈ ℕs))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2145  (class class class)co 7416   No csur 27874   1_s c1s 28069   +_s cadds 28222   -_s csubs 28283  ℕ_scnns 28576

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-ot 4596  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-2o 8459  df-nadd 8657  df-no 27877  df-lts 27878  df-bday 27879  df-les 27979  df-slts 28021  df-cuts 28023  df-0s 28070  df-1s 28071  df-made 28090  df-old 28091  df-left 28093  df-right 28094  df-norec 28201  df-norec2 28212  df-adds 28223  df-negs 28284  df-subs 28285  df-n0s 28577  df-nns 28578

This theorem is used by:  nnm1n0s  28638