Theorem peano2n0s 28230

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 28215	. . 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 27745	. . 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 28192	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕ0s)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ↦ cmpt 5191   “ cima 5644  (class class class)co 7390  ωcom 7845  reccrdg 8380   No csur 27558   0_s c0s 27741   1_s c1s 27742   +_s cadds 27873  ℕ_0scnn0s 28213

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  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 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-no 27561  df-slt 27562  df-bday 27563  df-sslt 27700  df-scut 27702  df-0s 27743  df-n0s 28215

This theorem is referenced by:  dfn0s2  28231  n0scut2  28234  n0addscl  28243  1n0s  28247  n0sfincut  28253  n0subs  28260  n0sleltp1  28263  bdayn0sf1o  28266  eucliddivs  28272  n0seo  28314  pw2cut  28342