Theorem peano2n0s 28252

Description: Peano postulate: the successor of a non-negative surreal integer is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
peano2n0s	⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕ0s)

Proof of Theorem peano2n0s

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-n0s 28237	. . 3 ⊢ ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ ℕ0s → ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω))
3		0sno 27763	. . 3 ⊢ 0s ∈ No
4	3	a1i 11	. 2 ⊢ (𝐴 ∈ ℕ0s → 0s ∈ No )
5		id 22	. 2 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ ℕ0s)
6	2, 4, 5	noseqp1 28214	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕ0s)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2110  Vcvv 3434   ↦ cmpt 5170   “ cima 5617  (class class class)co 7341  ωcom 7791  reccrdg 8323   No csur 27571   0_s c0s 27759   1_s c1s 27760   +_s cadds 27895  ℕ_0scnn0s 28235

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-10 2143  ax-11 2159  ax-12 2179  ax-ext 2702  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3344  df-reu 3345  df-rab 3394  df-v 3436  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4282  df-if 4474  df-pw 4550  df-sn 4575  df-pr 4577  df-tp 4579  df-op 4581  df-uni 4858  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6244  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6433  df-fun 6479  df-fn 6480  df-f 6481  df-f1 6482  df-fo 6483  df-f1o 6484  df-fv 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-no 27574  df-slt 27575  df-bday 27576  df-sslt 27714  df-scut 27716  df-0s 27761  df-n0s 28237

This theorem is referenced by:  dfn0s2  28253  n0scut2  28256  n0addscl  28265  1n0s  28269  n0sfincut  28275  n0subs  28282  n0sleltp1  28285  bdayn0sf1o  28288  eucliddivs  28294  n0seo  28337  pw2cut  28373  pw2cut2  28375  zs12half  28383  zs12zodd  28385