Theorem nn1m1nns 28386

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 2745	. . 3 ⊢ (𝑥 = 1s → (𝑥 = 1s ↔ 1s = 1s ))
2		oveq1 7366	. . . 4 ⊢ (𝑥 = 1s → (𝑥 -s 1s ) = ( 1s -s 1s ))
3	2	eleq1d 2826	. . 3 ⊢ (𝑥 = 1s → ((𝑥 -s 1s ) ∈ ℕs ↔ ( 1s -s 1s ) ∈ ℕs))
4	1, 3	orbi12d 925	. 2 ⊢ (𝑥 = 1s → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ ( 1s = 1s ∨ ( 1s -s 1s ) ∈ ℕs)))
5		eqeq1 2745	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1s ↔ 𝑦 = 1s ))
6		oveq1 7366	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 -s 1s ) = (𝑦 -s 1s ))
7	6	eleq1d 2826	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 -s 1s ) ∈ ℕs ↔ (𝑦 -s 1s ) ∈ ℕs))
8	5, 7	orbi12d 925	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ (𝑦 = 1s ∨ (𝑦 -s 1s ) ∈ ℕs)))
9		eqeq1 2745	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 = 1s ↔ (𝑦 +s 1s ) = 1s ))
10		oveq1 7366	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 1s ) = ((𝑦 +s 1s ) -s 1s ))
11	10	eleq1d 2826	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 -s 1s ) ∈ ℕs ↔ ((𝑦 +s 1s ) -s 1s ) ∈ ℕs))
12	9, 11	orbi12d 925	. 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ ((𝑦 +s 1s ) = 1s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕs)))
13		eqeq1 2745	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1s ↔ 𝐴 = 1s ))
14		oveq1 7366	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 -s 1s ) = (𝐴 -s 1s ))
15	14	eleq1d 2826	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 -s 1s ) ∈ ℕs ↔ (𝐴 -s 1s ) ∈ ℕs))
16	13, 15	orbi12d 925	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 1s ∨ (𝑥 -s 1s ) ∈ ℕs) ↔ (𝐴 = 1s ∨ (𝐴 -s 1s ) ∈ ℕs)))
17		eqid 2741	. . 3 ⊢ 1s = 1s
18	17	orci 872	. 2 ⊢ ( 1s = 1s ∨ ( 1s -s 1s ) ∈ ℕs)
19		nnno 28336	. . . . . 6 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No )
20		1no 27822	. . . . . 6 ⊢ 1s ∈ No
21		pncans 28084	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ 1s ∈ No ) → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
22	19, 20, 21	sylancl 593	. . . . 5 ⊢ (𝑦 ∈ ℕs → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
23		id 22	. . . . 5 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ ℕs)
24	22, 23	eqeltrd 2841	. . . 4 ⊢ (𝑦 ∈ ℕs → ((𝑦 +s 1s ) -s 1s ) ∈ ℕs)
25	24	olcd 881	. . 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 28385	1 ⊢ (𝐴 ∈ ℕs → (𝐴 = 1s ∨ (𝐴 -s 1s ) ∈ ℕs))

Syntax hints:   → wi 4   ∨ wo 854   = wceq 1548   ∈ wcel 2121  (class class class)co 7359   No csur 27624   1_s c1s 27818   +_s cadds 27971   -_s csubs 28032  ℕ_scnns 28325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rmo 3346  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3904  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-ot 4566  df-uni 4841  df-int 4880  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-tr 5182  df-id 5515  df-eprel 5520  df-po 5528  df-so 5529  df-fr 5573  df-se 5574  df-we 5575  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6255  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-riota 7316  df-ov 7362  df-oprab 7363  df-mpo 7364  df-om 7810  df-1st 7933  df-2nd 7934  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8343  df-1o 8399  df-2o 8400  df-nadd 8596  df-no 27627  df-lts 27628  df-bday 27629  df-les 27729  df-slts 27770  df-cuts 27772  df-0s 27819  df-1s 27820  df-made 27839  df-old 27840  df-left 27842  df-right 27843  df-norec 27950  df-norec2 27961  df-adds 27972  df-negs 28033  df-subs 28034  df-n0s 28326  df-nns 28327

This theorem is referenced by:  nnm1n0s  28387