Theorem peano5n0s 28315

Description: Peano's inductive postulate for non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
peano5n0s	⊢ (( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) → ℕ0s ⊆ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem peano5n0s

Dummy variables 𝑛 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-n0s 28310	. . 3 ⊢ ℕ0s = (rec((𝑛 ∈ V ↦ (𝑛 +s 1s )), 0s ) “ ω)
2	1	a1i 11	. 2 ⊢ (( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) → ℕ0s = (rec((𝑛 ∈ V ↦ (𝑛 +s 1s )), 0s ) “ ω))
3		0no 27805	. . 3 ⊢ 0s ∈ No
4	3	a1i 11	. 2 ⊢ (( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) → 0s ∈ No )
5		simpl 482	. 2 ⊢ (( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) → 0s ∈ 𝐴)
6		oveq1 7365	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
7	6	eleq1d 2821	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 +s 1s ) ∈ 𝐴 ↔ (𝑦 +s 1s ) ∈ 𝐴))
8	7	rspccva 3575	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦 +s 1s ) ∈ 𝐴)
9	8	adantll 714	. 2 ⊢ ((( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 +s 1s ) ∈ 𝐴)
10	2, 4, 5, 9	noseqind 28288	1 ⊢ (( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) → ℕ0s ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3051  Vcvv 3440   ⊆ wss 3901   ↦ cmpt 5179   “ cima 5627  (class class class)co 7358  ωcom 7808  reccrdg 8340   No csur 27607   0_s c0s 27801   1_s c1s 27802   +_s cadds 27955  ℕ_0scn0s 28308

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-tp 4585  df-op 4587  df-uni 4864  df-int 4903  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7809  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-rdg 8341  df-1o 8397  df-2o 8398  df-no 27610  df-lts 27611  df-bday 27612  df-slts 27754  df-cuts 27756  df-0s 27803  df-n0s 28310

This theorem is referenced by:  dfn0s2  28328