Theorem 0n0s 27937

Description: Peano postulate: 0_s is a non-negative surreal integer. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
0n0s	⊢ 0s ∈ ℕ0s

Proof of Theorem 0n0s

Step	Hyp	Ref	Expression
1		0sno 27562	. . . 4 ⊢ 0s ∈ No
2		fr0g 8440	. . . 4 ⊢ ( 0s ∈ No → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘∅) = 0s )
3	1, 2	ax-mp 5	. . 3 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘∅) = 0s
4		frfnom 8439	. . . 4 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω) Fn ω
5		peano1 7883	. . . 4 ⊢ ∅ ∈ ω
6		fnfvelrn 7083	. . . 4 ⊢ (((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω) Fn ω ∧ ∅ ∈ ω) → ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘∅) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω))
7	4, 5, 6	mp2an 688	. . 3 ⊢ ((rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)‘∅) ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)
8	3, 7	eqeltrri 2828	. 2 ⊢ 0s ∈ ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)
9		df-n0s 27929	. . 3 ⊢ ℕ0s = (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω)
10		df-ima 5690	. . 3 ⊢ (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) “ ω) = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)
11	9, 10	eqtri 2758	. 2 ⊢ ℕ0s = ran (rec((𝑥 ∈ V ↦ (𝑥 +s 1s )), 0s ) ↾ ω)
12	8, 11	eleqtrri 2830	1 ⊢ 0s ∈ ℕ0s

Syntax hints:   = wceq 1539   ∈ wcel 2104  Vcvv 3472  ∅c0 4323   ↦ cmpt 5232  ran crn 5678   ↾ cres 5679   “ cima 5680   Fn wfn 6539  ‘cfv 6544  (class class class)co 7413  ωcom 7859  reccrdg 8413   No csur 27377   0_s c0s 27558   1_s c1s 27559   +_s cadds 27679  ℕ_0scnn0s 27927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7729

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-2o 8471  df-no 27380  df-slt 27381  df-bday 27382  df-sslt 27517  df-scut 27519  df-0s 27560  df-n0s 27929

This theorem is referenced by:  dfn0s2  27939  n0sind  27940