Theorem n0p1nns 28575

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 7419	. . 3 ⊢ (𝑥 = 0s → (𝑥 +s 1s ) = ( 0s +s 1s ))
2	1	eleq1d 2847	. 2 ⊢ (𝑥 = 0s → ((𝑥 +s 1s ) ∈ ℕs ↔ ( 0s +s 1s ) ∈ ℕs))
3		oveq1 7419	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
4	3	eleq1d 2847	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 +s 1s ) ∈ ℕs ↔ (𝑦 +s 1s ) ∈ ℕs))
5		oveq1 7419	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 +s 1s ) = ((𝑦 +s 1s ) +s 1s ))
6	5	eleq1d 2847	. 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 +s 1s ) ∈ ℕs ↔ ((𝑦 +s 1s ) +s 1s ) ∈ ℕs))
7		oveq1 7419	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s 1s ) = (𝐴 +s 1s ))
8	7	eleq1d 2847	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 1s ) ∈ ℕs ↔ (𝐴 +s 1s ) ∈ ℕs))
9		1no 28014	. . . 4 ⊢ 1s ∈ No
10		addslid 28172	. . . 4 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
11	9, 10	ax-mp 5	. . 3 ⊢ ( 0s +s 1s ) = 1s
12		1nns 28553	. . 3 ⊢ 1s ∈ ℕs
13	11, 12	eqeltri 2858	. 2 ⊢ ( 0s +s 1s ) ∈ ℕs
14		peano2nns 28554	. . 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 28537	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2142  (class class class)co 7412   No csur 27815   0_s c0s 28009   1_s c1s 28010   +_s cadds 28163  ℕ_0scn0s 28516  ℕ_scnns 28517

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-ot 4597  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-se 5614  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-1o 8451  df-2o 8452  df-nadd 8650  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-0s 28011  df-1s 28012  df-made 28031  df-old 28032  df-left 28034  df-right 28035  df-norec2 28153  df-adds 28164  df-n0s 28518  df-nns 28519

This theorem is used by:  elzn0s  28602  bdayfinbndlem1  28671  z12zsodd  28686