Theorem n0p1nns 28361

Description: One plus a non-negative surreal integer is a positive surreal integer. (Contributed by Scott Fenton, 26-May-2025.)

Assertion

Ref	Expression
n0p1nns	⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)

Proof of Theorem n0p1nns

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

Step	Hyp	Ref	Expression
1		oveq1 7438	. . 3 ⊢ (𝑥 = 0s → (𝑥 +s 1s ) = ( 0s +s 1s ))
2	1	eleq1d 2826	. 2 ⊢ (𝑥 = 0s → ((𝑥 +s 1s ) ∈ ℕs ↔ ( 0s +s 1s ) ∈ ℕs))
3		oveq1 7438	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
4	3	eleq1d 2826	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 +s 1s ) ∈ ℕs ↔ (𝑦 +s 1s ) ∈ ℕs))
5		oveq1 7438	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 +s 1s ) = ((𝑦 +s 1s ) +s 1s ))
6	5	eleq1d 2826	. 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 +s 1s ) ∈ ℕs ↔ ((𝑦 +s 1s ) +s 1s ) ∈ ℕs))
7		oveq1 7438	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s 1s ) = (𝐴 +s 1s ))
8	7	eleq1d 2826	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 1s ) ∈ ℕs ↔ (𝐴 +s 1s ) ∈ ℕs))
9		1sno 27872	. . . 4 ⊢ 1s ∈ No
10		addslid 28001	. . . 4 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
11	9, 10	ax-mp 5	. . 3 ⊢ ( 0s +s 1s ) = 1s
12		1nns 28352	. . 3 ⊢ 1s ∈ ℕs
13	11, 12	eqeltri 2837	. 2 ⊢ ( 0s +s 1s ) ∈ ℕs
14		peano2nns 28353	. . 3 ⊢ ((𝑦 +s 1s ) ∈ ℕs → ((𝑦 +s 1s ) +s 1s ) ∈ ℕs)
15	14	a1i 11	. 2 ⊢ (𝑦 ∈ ℕ0s → ((𝑦 +s 1s ) ∈ ℕs → ((𝑦 +s 1s ) +s 1s ) ∈ ℕs))
16	2, 4, 6, 8, 13, 15	n0sind 28337	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  (class class class)co 7431   No csur 27684   0_s c0s 27867   1_s c1s 27868   +_s cadds 27992  ℕ_0scnn0s 28318  ℕ_scnns 28319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-1s 27870  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec2 27982  df-adds 27993  df-n0s 28320  df-nns 28321

This theorem is referenced by:  elzn0s  28384