Theorem peano2n0s 28481

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 28465	. . 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 27960	. . 3 ⊢ 0s ∈ No
4	3	a1i 11	. 2 ⊢ (𝐴 ∈ ℕ0s → 0s ∈ No )
5		id 23	. 2 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ ℕ0s)
6	2, 4, 5	noseqp1 28442	1 ⊢ (𝐴 ∈ ℕ0s → (𝐴 +s 1s ) ∈ ℕ0s)

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ↦ cmpt 5186   “ cima 5655  (class class class)co 7400  ωcom 7850  reccrdg 8384   No csur 27762   0_s c0s 27956   1_s c1s 27957   +_s cadds 28110  ℕ_0scn0s 28463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-no 27765  df-lts 27766  df-bday 27767  df-slts 27909  df-cuts 27911  df-0s 27958  df-n0s 28465

This theorem is referenced by:  peano2n0sd  28482  dfn0s2  28483  n0cut2  28486  n0addscl  28495  1n0s  28499  n0subs  28514  n0lesltp1  28517  bdayn0sf1o  28521  eucliddivs  28527  n0seo  28572  pw2cut  28611  pw2cut2  28613  bdaypw2n0bndlem  28614  bdaypw2bnd  28616  z12shalf  28631  z12zsodd  28633