Theorem nn1m1nns 28270

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 2734	. . 3 ⊢ (𝑥 = 1s → (𝑥 = 1s ↔ 1s = 1s ))
2		oveq1 7397	. . . 4 ⊢ (𝑥 = 1s → (𝑥 -s 1s ) = ( 1s -s 1s ))
3	2	eleq1d 2814	. . 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 2734	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1s ↔ 𝑦 = 1s ))
6		oveq1 7397	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 -s 1s ) = (𝑦 -s 1s ))
7	6	eleq1d 2814	. . 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 2734	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 = 1s ↔ (𝑦 +s 1s ) = 1s ))
10		oveq1 7397	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 1s ) = ((𝑦 +s 1s ) -s 1s ))
11	10	eleq1d 2814	. . 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 2734	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1s ↔ 𝐴 = 1s ))
14		oveq1 7397	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 -s 1s ) = (𝐴 -s 1s ))
15	14	eleq1d 2814	. . 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 2730	. . 3 ⊢ 1s = 1s
18	17	orci 865	. 2 ⊢ ( 1s = 1s ∨ ( 1s -s 1s ) ∈ ℕs)
19		nnsno 28224	. . . . . 6 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No )
20		1sno 27746	. . . . . 6 ⊢ 1s ∈ No
21		pncans 27983	. . . . . 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 2829	. . . 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 28269	1 ⊢ (𝐴 ∈ ℕs → (𝐴 = 1s ∨ (𝐴 -s 1s ) ∈ ℕs))

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1540   ∈ wcel 2109  (class class class)co 7390   No csur 27558   1_s c1s 27742   +_s cadds 27873   -_s csubs 27933  ℕ_scnns 28214

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 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-ot 4601  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-nadd 8633  df-no 27561  df-slt 27562  df-bday 27563  df-sle 27664  df-sslt 27700  df-scut 27702  df-0s 27743  df-1s 27744  df-made 27762  df-old 27763  df-left 27765  df-right 27766  df-norec 27852  df-norec2 27863  df-adds 27874  df-negs 27934  df-subs 27935  df-n0s 28215  df-nns 28216

This theorem is referenced by:  nnm1n0s  28271