Theorem n0sexg 28759

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 28757	. 2 ⊢ ℕ0s = (rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) “ ω)
2		rdgfun 8462	. . 3 ⊢ Fun rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s )
3		funimaexg 6656	. . 3 ⊢ ((Fun rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) ∧ ω ∈ V) → (rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) “ ω) ∈ V)
4	2, 3	mpan 703	. 2 ⊢ (ω ∈ V → (rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) “ ω) ∈ V)
5	1, 4	eqeltrid 2868	1 ⊢ (ω ∈ V → ℕ0s ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2147  Vcvv 3454   ↦ cmpt 5192   “ cima 5661  Fun wfun 6560  (class class class)co 7451  ωcom 7911  reccrdg 8455   0_s c0s 28248   1_s c1s 28249   +_s cadds 28402  ℕ_0scn0s 28755

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-n0s 28757

This theorem is used by:  n0sex  28760  oldfib  28820