Theorem n0s0m1 28805

Description: Every non-negative surreal integer is either zero or a successor. (Contributed by Scott Fenton, 26-May-2025.)

Assertion

Ref	Expression
n0s0m1	⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 −s 1s ) ∈ ℕ0s))

Proof of Theorem n0s0m1

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2767	. . 3 ⊢ (𝑥 = 0s → (𝑥 = 0s ↔ 0s = 0s ))
2		oveq1 7458	. . . 4 ⊢ (𝑥 = 0s → (𝑥 −s 1s ) = ( 0s −s 1s ))
3	2	eleq1d 2849	. . 3 ⊢ (𝑥 = 0s → ((𝑥 −s 1s ) ∈ ℕ0s ↔ ( 0s −s 1s ) ∈ ℕ0s))
4	1, 3	orbi12d 932	. 2 ⊢ (𝑥 = 0s → ((𝑥 = 0s ∨ (𝑥 −s 1s ) ∈ ℕ0s) ↔ ( 0s = 0s ∨ ( 0s −s 1s ) ∈ ℕ0s)))
5		eqeq1 2767	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 0s ↔ 𝑦 = 0s ))
6		oveq1 7458	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 −s 1s ) = (𝑦 −s 1s ))
7	6	eleq1d 2849	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 −s 1s ) ∈ ℕ0s ↔ (𝑦 −s 1s ) ∈ ℕ0s))
8	5, 7	orbi12d 932	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 0s ∨ (𝑥 −s 1s ) ∈ ℕ0s) ↔ (𝑦 = 0s ∨ (𝑦 −s 1s ) ∈ ℕ0s)))
9		eqeq1 2767	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 = 0s ↔ (𝑦 +s 1s ) = 0s ))
10		oveq1 7458	. . . 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 ) ∈ ℕ0s ↔ ((𝑦 +s 1s ) −s 1s ) ∈ ℕ0s))
12	9, 11	orbi12d 932	. 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 = 0s ∨ (𝑥 −s 1s ) ∈ ℕ0s) ↔ ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) −s 1s ) ∈ ℕ0s)))
13		eqeq1 2767	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 0s ↔ 𝐴 = 0s ))
14		oveq1 7458	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 −s 1s ) = (𝐴 −s 1s ))
15	14	eleq1d 2849	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 −s 1s ) ∈ ℕ0s ↔ (𝐴 −s 1s ) ∈ ℕ0s))
16	13, 15	orbi12d 932	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 0s ∨ (𝑥 −s 1s ) ∈ ℕ0s) ↔ (𝐴 = 0s ∨ (𝐴 −s 1s ) ∈ ℕ0s)))
17		eqid 2763	. . 3 ⊢ 0s = 0s
18	17	orci 879	. 2 ⊢ ( 0s = 0s ∨ ( 0s −s 1s ) ∈ ℕ0s)
19		n0no 28766	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No)
20		1no 28253	. . . . . 6 ⊢ 1s ∈ No
21		pncans 28515	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ 1s ∈ No) → ((𝑦 +s 1s ) −s 1s ) = 𝑦)
22	19, 20, 21	sylancl 598	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) −s 1s ) = 𝑦)
23		id 23	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℕ0s)
24	22, 23	eqeltrd 2864	. . . 4 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) −s 1s ) ∈ ℕ0s)
25	24	olcd 888	. . 3 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) −s 1s ) ∈ ℕ0s))
26	25	a1d 26	. 2 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 = 0s ∨ (𝑦 −s 1s ) ∈ ℕ0s) → ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) −s 1s ) ∈ ℕ0s)))
27	4, 8, 12, 16, 18, 26	n0sind 28776	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 −s 1s ) ∈ ℕ0s))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1570   ∈ wcel 2147  (class class class)co 7451  Nocsur 28054   0_s c0s 28248   1_s c1s 28249   +_s cadds 28402   −_s csubs 28463  ℕ_0scn0s 28755

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 4841  df-ot 4864  df-uni 4873  df-int 4913  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-2o 8515  df-nadd 8713  df-no 28057  df-lts 28058  df-bday 28059  df-les 28159  df-slts 28201  df-cuts 28203  df-0s 28250  df-1s 28251  df-made 28270  df-old 28271  df-left 28273  df-right 28274  df-norec 28381  df-norec2 28392  df-adds 28403  df-negs 28464  df-subs 28465  df-n0s 28757

This theorem is used by:  n0subs  28806