Theorem n0p1nns 28466

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 7405	. . 3 ⊢ (𝑥 = 0s → (𝑥 +s 1s ) = ( 0s +s 1s ))
2	1	eleq1d 2849	. 2 ⊢ (𝑥 = 0s → ((𝑥 +s 1s ) ∈ ℕs ↔ ( 0s +s 1s ) ∈ ℕs))
3		oveq1 7405	. . 3 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
4	3	eleq1d 2849	. 2 ⊢ (𝑥 = 𝑦 → ((𝑥 +s 1s ) ∈ ℕs ↔ (𝑦 +s 1s ) ∈ ℕs))
5		oveq1 7405	. . 3 ⊢ (𝑥 = (𝑦 +s 1s ) → (𝑥 +s 1s ) = ((𝑦 +s 1s ) +s 1s ))
6	5	eleq1d 2849	. 2 ⊢ (𝑥 = (𝑦 +s 1s ) → ((𝑥 +s 1s ) ∈ ℕs ↔ ((𝑦 +s 1s ) +s 1s ) ∈ ℕs))
7		oveq1 7405	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 +s 1s ) = (𝐴 +s 1s ))
8	7	eleq1d 2849	. 2 ⊢ (𝑥 = 𝐴 → ((𝑥 +s 1s ) ∈ ℕs ↔ (𝐴 +s 1s ) ∈ ℕs))
9		1no 27905	. . . 4 ⊢ 1s ∈ No
10		addslid 28063	. . . 4 ⊢ ( 1s ∈ No → ( 0s +s 1s ) = 1s )
11	9, 10	ax-mp 5	. . 3 ⊢ ( 0s +s 1s ) = 1s
12		1nns 28444	. . 3 ⊢ 1s ∈ ℕs
13	11, 12	eqeltri 2860	. 2 ⊢ ( 0s +s 1s ) ∈ ℕs
14		peano2nns 28445	. . 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 28428	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕs)

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144  (class class class)co 7398   No csur 27706   0_s c0s 27900   1_s c1s 27901   +_s cadds 28054  ℕ_0scn0s 28407  ℕ_scnns 28408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-ot 4593  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-se 5603  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-pred 6290  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-riota 7355  df-ov 7401  df-oprab 7402  df-mpo 7403  df-om 7849  df-1st 7972  df-2nd 7973  df-frecs 8264  df-wrecs 8295  df-recs 8344  df-rdg 8383  df-1o 8439  df-2o 8440  df-nadd 8638  df-no 27709  df-lts 27710  df-bday 27711  df-les 27811  df-slts 27853  df-cuts 27855  df-0s 27902  df-1s 27903  df-made 27922  df-old 27923  df-left 27925  df-right 27926  df-norec2 28044  df-adds 28055  df-n0s 28409  df-nns 28410

This theorem is referenced by:  elzn0s  28493  bdayfinbndlem1  28562  z12zsodd  28577