Theorem n0sexg 28277

Description: The set of all non-negative surreal integers exists. This theorem avoids the axiom of infinity by including it as an antecedent. (Contributed by Scott Fenton, 20-Feb-2025.)

Assertion

Ref	Expression
n0sexg	⊢ (ω ∈ V → ℕ0s ∈ V)

Proof of Theorem n0sexg

Step	Hyp	Ref	Expression
1		df-n0s 28275	. 2 ⊢ ℕ0s = (rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) “ ω)
2		rdgfun 8345	. . 3 ⊢ Fun rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s )
3		funimaexg 6577	. . 3 ⊢ ((Fun rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) ∧ ω ∈ V) → (rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) “ ω) ∈ V)
4	2, 3	mpan 690	. 2 ⊢ (ω ∈ V → (rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) “ ω) ∈ V)
5	1, 4	eqeltrid 2838	1 ⊢ (ω ∈ V → ℕ0s ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2113  Vcvv 3438   ↦ cmpt 5177   “ cima 5625  Fun wfun 6484  (class class class)co 7356  ωcom 7806  reccrdg 8338   0_s c0s 27793   1_s c1s 27794   +_s cadds 27929  ℕ_0scnn0s 28273

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

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 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-n0s 28275

This theorem is referenced by:  n0sex  28278  oldfib  28335