Theorem n0s0m1 28377

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 7455	. . . 4 ⊢ (𝑥 = 0s → (𝑥 -s 1s ) = ( 0s -s 1s ))
3	2	eleq1d 2829	. . 3 ⊢ (𝑥 = 0s → ((𝑥 -s 1s ) ∈ ℕ0s ↔ ( 0s -s 1s ) ∈ ℕ0s))
4	1, 3	orbi12d 917	. 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 7455	. . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 -s 1s ) = (𝑦 -s 1s ))
7	6	eleq1d 2829	. . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 -s 1s ) ∈ ℕ0s ↔ (𝑦 -s 1s ) ∈ ℕ0s))
8	5, 7	orbi12d 917	. 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 7455	. . . 4 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 -s 1s ) = ((𝑦 +s 1s ) -s 1s ))
11	10	eleq1d 2829	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 -s 1s ) ∈ ℕ0s ↔ ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s))
12	9, 11	orbi12d 917	. 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 7455	. . . 4 ⊢ (𝑥 = 𝐴 → (𝑥 -s 1s ) = (𝐴 -s 1s ))
15	14	eleq1d 2829	. . 3 ⊢ (𝑥 = 𝐴 → ((𝑥 -s 1s ) ∈ ℕ0s ↔ (𝐴 -s 1s ) ∈ ℕ0s))
16	13, 15	orbi12d 917	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 = 0s ∨ (𝑥 -s 1s ) ∈ ℕ0s) ↔ (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s)))
17		eqid 2740	. . 3 ⊢ 0s = 0s
18	17	orci 864	. 2 ⊢ ( 0s = 0s ∨ ( 0s -s 1s ) ∈ ℕ0s)
19		n0sno 28346	. . . . . 6 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ No )
20		1sno 27890	. . . . . 6 ⊢ 1s ∈ No
21		pncans 28120	. . . . . 6 ⊢ ((𝑦 ∈ No ∧ 1s ∈ No ) → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
22	19, 20, 21	sylancl 585	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 1s ) = 𝑦)
23		id 22	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → 𝑦 ∈ ℕ0s)
24	22, 23	eqeltrd 2844	. . . 4 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) -s 1s ) ∈ ℕ0s)
25	24	olcd 873	. . 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 28355	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 = 0s ∨ (𝐴 -s 1s ) ∈ ℕ0s))

Syntax hints:   → wi 4   ∨ wo 846   = wceq 1537   ∈ wcel 2108  (class class class)co 7448   No csur 27702   0_s c0s 27885   1_s c1s 27886   +_s cadds 28010   -_s csubs 28070  ℕ_0scnn0s 28336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-ot 4657  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-nadd 8722  df-no 27705  df-slt 27706  df-bday 27707  df-sle 27808  df-sslt 27844  df-scut 27846  df-0s 27887  df-1s 27888  df-made 27904  df-old 27905  df-left 27907  df-right 27908  df-norec 27989  df-norec2 28000  df-adds 28011  df-negs 28071  df-subs 28072  df-n0s 28338

This theorem is referenced by:  n0subs  28378