Theorem n0sexg 28714

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 28712	. 2 ⊢ ℕ0s = (rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) “ ω)
2		rdgfun 8435	. . 3 ⊢ Fun rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s )
3		funimaexg 6634	. . 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 2867	1 ⊢ (ω ∈ V → ℕ0s ∈ V)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146  Vcvv 3453   ↦ cmpt 5188   “ cima 5657  Fun wfun 6540  (class class class)co 7427  ωcom 7884  reccrdg 8428   0_s c0s 28203   1_s c1s 28204   +_s cadds 28357  ℕ_0scn0s 28710

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5234  ax-sep 5251  ax-nul 5262  ax-pr 5394  ax-un 7758

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3369  df-rab 3416  df-v 3455  df-sbc 3742  df-csb 3851  df-ss 3900  df-pss 4016  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-pw 4561  df-sn 4588  df-pr 4590  df-op 4838  df-uni 4870  df-iun 4955  df-br 5106  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5549  df-eprel 5554  df-po 5562  df-so 5563  df-fr 5607  df-we 5609  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6310  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6500  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7430  df-2nd 8009  df-frecs 8310  df-wrecs 8341  df-recs 8390  df-rdg 8429  df-n0s 28712

This theorem is used by:  n0sex  28715  oldfib  28775