Theorem nn1m1nns 28304

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 2733	. . 3 ⊢ (𝑥 = 1s → (𝑥 = 1s ↔ 1s = 1s ))
2		oveq1 7376	. . . 4 ⊢ (𝑥 = 1s → (𝑥 -s 1s ) = ( 1s -s 1s ))
3	2	eleq1d 2813	. . 3 ⊢ (𝑥 = 1s → ((𝑥 -s 1s ) ∈ ℕs ↔ ( 1s -s 1s ) ∈ ℕs))
4	1, 3	orbi12d 918	. 2 ⊢ (𝑥 = 1s → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ ( 1s = 1s ∨ ( 1s -s 1s ) ∈ ℕs)))
5		eqeq1 2733	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1s ↔ 𝑦 = 1s ))
6		oveq1 7376	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 -s 1s ) = (𝑦 -s 1s ))
7	6	eleq1d 2813	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 -s 1s ) ∈ ℕs ↔ (𝑦 -s 1s ) ∈ ℕs))
8	5, 7	orbi12d 918	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ (𝑦 = 1s ∨ (𝑦 -s 1s ) ∈ ℕs)))
9		eqeq1 2733	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 = 1s ↔ (𝑦 +s 1s ) = 1s ))
10		oveq1 7376	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 1s ) = ((𝑦 +s 1s ) -s 1s ))
11	10	eleq1d 2813	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 -s 1s ) ∈ ℕs ↔ ((𝑦 +s 1s ) -s 1s ) ∈ ℕs))
12	9, 11	orbi12d 918	. 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ ((𝑦 +s 1s ) = 1s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕs)))
13		eqeq1 2733	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1s ↔ 𝐴 = 1s ))
14		oveq1 7376	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 -s 1s ) = (𝐴 -s 1s ))
15	14	eleq1d 2813	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 -s 1s ) ∈ ℕs ↔ (𝐴 -s 1s ) ∈ ℕs))
16	13, 15	orbi12d 918	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ (𝐴 = 1s ∨ (𝐴 -s 1s ) ∈ ℕs)))
17		eqid 2729	. . 3 ⊢ 1s = 1s
18	17	orci 865	. 2 ⊢ ( 1s = 1s ∨ ( 1s -s 1s ) ∈ ℕs)
19		nnsno 28258	. . . . . 6 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No )
20		1sno 27777	. . . . . 6 ⊢ 1s ∈ No
21		pncans 28017	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ 1s ∈ No ) → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
22	19, 20, 21	sylancl 586	. . . . 5 ⊢ (𝑦 ∈ ℕs → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
23		id 22	. . . . 5 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ ℕs)
24	22, 23	eqeltrd 2828	. . . 4 ⊢ (𝑦 ∈ ℕs → ((𝑦 +s 1s ) -s 1s ) ∈ ℕs)
25	24	olcd 874	. . 3 ⊢ (𝑦 ∈ ℕs → ((𝑦 +s 1s ) = 1s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕs))
26	25	a1d 25	. 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 28303	1 ⊢ (𝐴 ∈ ℕs → (𝐴 = 1s ∨ (𝐴 -s 1s ) ∈ ℕs))

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1540   ∈ wcel 2109  (class class class)co 7369   No csur 27585   1_s c1s 27773   +_s cadds 27907   -_s csubs 27967  ℕ_scnns 28248

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-ot 4594  df-uni 4868  df-int 4907  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-1o 8411  df-2o 8412  df-nadd 8607  df-no 27588  df-slt 27589  df-bday 27590  df-sle 27691  df-sslt 27728  df-scut 27730  df-0s 27774  df-1s 27775  df-made 27793  df-old 27794  df-left 27796  df-right 27797  df-norec 27886  df-norec2 27897  df-adds 27908  df-negs 27968  df-subs 27969  df-n0s 28249  df-nns 28250

This theorem is referenced by:  nnm1n0s  28305