Theorem nn1m1nns 28578

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 7419	. . . 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 931	. 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 7419	. . . 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 931	. 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 7419	. . . 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 931	. 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 7419	. . . 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 931	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ (𝐴 = 1s ∨ (𝐴 -s 1s ) ∈ ℕs)))
17		eqid 2762	. . 3 ⊢ 1s = 1s
18	17	orci 878	. 2 ⊢ ( 1s = 1s ∨ ( 1s -s 1s ) ∈ ℕs)
19		nnno 28528	. . . . . 6 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No )
20		1no 28014	. . . . . 6 ⊢ 1s ∈ No
21		pncans 28276	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ 1s ∈ No ) → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
22	19, 20, 21	sylancl 597	. . . . 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 887	. . 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 28577	1 ⊢ (𝐴 ∈ ℕs → (𝐴 = 1s ∨ (𝐴 -s 1s ) ∈ ℕs))

This proof depends on syntax axioms:   → wi 4   ∨ wo 860   = wceq 1569   ∈ wcel 2142  (class class class)co 7412   No csur 27815   1_s c1s 28010   +_s cadds 28163   -_s csubs 28224  ℕ_scnns 28517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-nadd 8650  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-0s 28011  df-1s 28012  df-made 28031  df-old 28032  df-left 28034  df-right 28035  df-norec 28142  df-norec2 28153  df-adds 28164  df-negs 28225  df-subs 28226  df-n0s 28518  df-nns 28519

This theorem is used by:  nnm1n0s  28579