Theorem n0sexg 28475

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 28473	. 2 ⊢ ℕ0s = (rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) “ ω)
2		rdgfun 8403	. . 3 ⊢ Fun rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s )
3		funimaexg 6623	. . 3 ⊢ ((Fun rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) ∧ ω ∈ V) → (rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) “ ω) ∈ V)
4	2, 3	mpan 702	. 2 ⊢ (ω ∈ V → (rec((𝑓 ∈ V ↦ (𝑓 +s 1s )), 0s ) “ ω) ∈ V)
5	1, 4	eqeltrid 2873	1 ⊢ (ω ∈ V → ℕ0s ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2149  Vcvv 3463   ↦ cmpt 5196   “ cima 5665  Fun wfun 6531  (class class class)co 7411  ωcom 7862  reccrdg 8396   0_s c0s 27964   1_s c1s 27965   +_s cadds 28118  ℕ_0scn0s 28471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-n0s 28473

This theorem is referenced by:  n0sex  28476  oldfib  28536