Theorem nn1m1nns 28790

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 2767	. . 3 ⊢ (𝑥 = 1s → (𝑥 = 1s ↔ 1s = 1s ))
2		oveq1 7450	. . . 4 ⊢ (𝑥 = 1s → (𝑥 −s 1s ) = ( 1s −s 1s ))
3	2	eleq1d 2849	. . 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 2767	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 1s ↔ 𝑦 = 1s ))
6		oveq1 7450	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 −s 1s ) = (𝑦 −s 1s ))
7	6	eleq1d 2849	. . 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 2767	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 = 1s ↔ (𝑦 +s 1s ) = 1s ))
10		oveq1 7450	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 −s 1s ) = ((𝑦 +s 1s ) −s 1s ))
11	10	eleq1d 2849	. . 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 2767	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 1s ↔ 𝐴 = 1s ))
14		oveq1 7450	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 −s 1s ) = (𝐴 −s 1s ))
15	14	eleq1d 2849	. . 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 2763	. . 3 ⊢ 1s = 1s
18	17	orci 879	. 2 ⊢ ( 1s = 1s ∨ ( 1s −s 1s ) ∈ ℕs)
19		nnno 28740	. . . . . 6 ⊢ (𝑦 ∈ ℕs → 𝑦 ∈ No)
20		1no 28226	. . . . . 6 ⊢ 1s ∈ No
21		pncans 28488	. . . . . 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 2864	. . . 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 28789	1 ⊢ (𝐴 ∈ ℕs → (𝐴 = 1s ∨ (𝐴 −s 1s ) ∈ ℕs))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2147  (class class class)co 7443  Nocsur 28027   1_s c1s 28222   +_s cadds 28375   −_s csubs 28436  ℕ_scnns 28729

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-lim 6385  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-om 7902  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-rdg 8446  df-1o 8504  df-2o 8505  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-les 28132  df-slts 28174  df-cuts 28176  df-0s 28223  df-1s 28224  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-subs 28438  df-n0s 28730  df-nns 28731

This theorem is used by:  nnm1n0s  28791