Theorem n0s0m1 28374

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 2739	. . 3 ⊢ (𝑥 = 0s → (𝑥 = 0s ↔ 0s = 0s ))
2		oveq1 7438	. . . 4 ⊢ (𝑥 = 0s → (𝑥 -s 1s ) = ( 0s -s 1s ))
3	2	eleq1d 2824	. . 3 ⊢ (𝑥 = 0s → ((𝑥 -s 1s ) ∈ ℕ0s ↔ ( 0s -s 1s ) ∈ ℕ0s))
4	1, 3	orbi12d 918	. 2 ⊢ (𝑥 = 0s → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ ( 0s = 0s ∨ ( 0s -s 1s ) ∈ ℕ0s)))
5		eqeq1 2739	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 0s ↔ 𝑦 = 0s ))
6		oveq1 7438	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 -s 1s ) = (𝑦 -s 1s ))
7	6	eleq1d 2824	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 -s 1s ) ∈ ℕ0s ↔ (𝑦 -s 1s ) ∈ ℕ0s))
8	5, 7	orbi12d 918	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ (𝑦 = 0s ∨ (𝑦 -s 1s ) ∈ ℕ0s)))
9		eqeq1 2739	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 = 0s ↔ (𝑦 +s 1s ) = 0s ))
10		oveq1 7438	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 1s ) = ((𝑦 +s 1s ) -s 1s ))
11	10	eleq1d 2824	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 -s 1s ) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s))
12	9, 11	orbi12d 918	. 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s)))
13		eqeq1 2739	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 0s ↔ 𝐴 = 0s ))
14		oveq1 7438	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 -s 1s ) = (𝐴 -s 1s ))
15	14	eleq1d 2824	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 -s 1s ) ∈ ℕ0s ↔ (𝐴 -s 1s ) ∈ ℕ0s))
16	13, 15	orbi12d 918	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s)))
17		eqid 2735	. . 3 ⊢ 0s = 0s
18	17	orci 865	. 2 ⊢ ( 0s = 0s ∨ ( 0s -s 1s ) ∈ ℕ0s)
19		n0sno 28343	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No )
20		1sno 27887	. . . . . 6 ⊢ 1s ∈ No
21		pncans 28117	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ 1s ∈ No ) → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
22	19, 20, 21	sylancl 586	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
23		id 22	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℕ0s)
24	22, 23	eqeltrd 2839	. . . 4 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s)
25	24	olcd 874	. . 3 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s))
26	25	a1d 25	. 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 28352	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s))

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1537   ∈ wcel 2106  (class class class)co 7431   No csur 27699   0_s c0s 27882   1_s c1s 27883   +_s cadds 28007   -_s csubs 28067  ℕ_0scnn0s 28333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-1s 27885  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068  df-subs 28069  df-n0s 28335

This theorem is referenced by:  n0subs  28375