Theorem peano2n0s 28393

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 28377	. . 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		0no 27872	. . 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 28354	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕ0s)

Syntax hints:   → wi 4   = wceq 1554   ∈ wcel 2136  Vcvv 3448   ↦ cmpt 5175   “ cima 5643  (class class class)co 7385  ωcom 7835  reccrdg 8368   No csur 27674   0_s c0s 27868   1_s c1s 27869   +_s cadds 28022  ℕ_0scn0s 28375

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728  ax-rep 5221  ax-sep 5240  ax-nul 5250  ax-pow 5316  ax-pr 5384  ax-un 7707

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3or 1096  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-eu 2590  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-rex 3081  df-rmo 3361  df-reu 3362  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4281  df-if 4475  df-pw 4551  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4900  df-iun 4945  df-br 5095  df-opab 5157  df-mpt 5176  df-tr 5202  df-id 5535  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6466  df-fun 6512  df-fn 6513  df-f 6514  df-f1 6515  df-fo 6516  df-f1o 6517  df-fv 6518  df-riota 7342  df-ov 7388  df-oprab 7389  df-mpo 7390  df-om 7836  df-2nd 7960  df-frecs 8250  df-wrecs 8281  df-recs 8330  df-rdg 8369  df-1o 8425  df-2o 8426  df-no 27677  df-lts 27678  df-bday 27679  df-slts 27821  df-cuts 27823  df-0s 27870  df-n0s 28377

This theorem is referenced by:  peano2n0sd  28394  dfn0s2  28395  n0cut2  28398  n0addscl  28407  1n0s  28411  n0subs  28426  n0lesltp1  28429  bdayn0sf1o  28433  eucliddivs  28439  n0seo  28484  pw2cut  28523  pw2cut2  28525  bdaypw2n0bndlem  28526  bdaypw2bnd  28528  z12shalf  28543  z12zsodd  28545