Theorem n0p1nns 28757

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 7427	. . 3 ⊢ (𝑥 = 0s → (𝑥 +s 1s ) = ( 0s +s 1s ))
2	1	eleq1d 2845	. 2 ⊢ (𝑥 = 0s → ((𝑥 +s 1s ) ∈ ℕs ↔ ( 0s +s 1s ) ∈ ℕs))
3		oveq1 7427	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
4	3	eleq1d 2845	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 +s 1s ) ∈ ℕs ↔ (𝑦 +s 1s ) ∈ ℕs))
5		oveq1 7427	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 +s 1s ) = ((𝑦 +s 1s ) +s 1s ))
6	5	eleq1d 2845	. 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 +s 1s ) ∈ ℕs ↔ ((𝑦 +s 1s ) +s 1s ) ∈ ℕs))
7		oveq1 7427	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s 1s ) = (𝐴 +s 1s ))
8	7	eleq1d 2845	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 1s ) ∈ ℕs ↔ (𝐴 +s 1s ) ∈ ℕs))
9		1no 28196	. . . 4 ⊢ 1s ∈ No
10		addslid 28354	. . . 4 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
11	9, 10	ax-mp 5	. . 3 ⊢ ( 0s +s 1s ) = 1s
12		1nns 28735	. . 3 ⊢ 1s ∈ ℕs
13	11, 12	eqeltri 2856	. 2 ⊢ ( 0s +s 1s ) ∈ ℕs
14		peano2nns 28736	. . 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 28719	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  (class class class)co 7420  Nocsur 27997   0_s c0s 28191   1_s c1s 28192   +_s cadds 28345  ℕ_0scn0s 28698  ℕ_scnns 28699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-ot 4592  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-nadd 8675  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-0s 28193  df-1s 28194  df-made 28213  df-old 28214  df-left 28216  df-right 28217  df-norec2 28335  df-adds 28346  df-n0s 28700  df-nns 28701

This theorem is used by:  elzn0s  28784  bdayfinbndlem1  28853  z12zsodd  28868