Theorem n0s0m1 28379

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 2744	. . 3 ⊢ (𝑥 = 0s → (𝑥 = 0s ↔ 0s = 0s ))
2		oveq1 7370	. . . 4 ⊢ (𝑥 = 0s → (𝑥 -s 1s ) = ( 0s -s 1s ))
3	2	eleq1d 2825	. . 3 ⊢ (𝑥 = 0s → ((𝑥 -s 1s ) ∈ ℕ0s ↔ ( 0s -s 1s ) ∈ ℕ0s))
4	1, 3	orbi12d 924	. 2 ⊢ (𝑥 = 0s → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ ( 0s = 0s ∨ ( 0s -s 1s ) ∈ ℕ0s)))
5		eqeq1 2744	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 = 0s ↔ 𝑦 = 0s ))
6		oveq1 7370	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 -s 1s ) = (𝑦 -s 1s ))
7	6	eleq1d 2825	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 -s 1s ) ∈ ℕ0s ↔ (𝑦 -s 1s ) ∈ ℕ0s))
8	5, 7	orbi12d 924	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ (𝑦 = 0s ∨ (𝑦 -s 1s ) ∈ ℕ0s)))
9		eqeq1 2744	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 = 0s ↔ (𝑦 +s 1s ) = 0s ))
10		oveq1 7370	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 1s ) = ((𝑦 +s 1s ) -s 1s ))
11	10	eleq1d 2825	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 -s 1s ) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s))
12	9, 11	orbi12d 924	. 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ ((𝑦 +s 1s ) = 0s ∨ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s)))
13		eqeq1 2744	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 = 0s ↔ 𝐴 = 0s ))
14		oveq1 7370	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 -s 1s ) = (𝐴 -s 1s ))
15	14	eleq1d 2825	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 -s 1s ) ∈ ℕ0s ↔ (𝐴 -s 1s ) ∈ ℕ0s))
16	13, 15	orbi12d 924	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s)))
17		eqid 2740	. . 3 ⊢ 0s = 0s
18	17	orci 871	. 2 ⊢ ( 0s = 0s ∨ ( 0s -s 1s ) ∈ ℕ0s)
19		n0no 28340	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No )
20		1no 27827	. . . . . 6 ⊢ 1s ∈ No
21		pncans 28089	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ 1s ∈ No ) → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
22	19, 20, 21	sylancl 592	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
23		id 22	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℕ0s)
24	22, 23	eqeltrd 2840	. . . 4 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s)
25	24	olcd 880	. . 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 28350	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s))

Syntax hints:   → wi 4   ∨ wo 853   = wceq 1547   ∈ wcel 2119  (class class class)co 7363   No csur 27628   0_s c0s 27822   1_s c1s 27823   +_s cadds 27976   -_s csubs 28037  ℕ_0scn0s 28329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-ot 4571  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-om 7814  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-rdg 8346  df-1o 8402  df-2o 8403  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-les 27734  df-slts 27775  df-cuts 27777  df-0s 27824  df-1s 27825  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038  df-subs 28039  df-n0s 28331

This theorem is referenced by:  n0subs  28380