Theorem peano5n0s 28342

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 28338	. . 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		0sno 27889	. . 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 7455	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
7	6	eleq1d 2829	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 +s 1s ) ∈ 𝐴 ↔ (𝑦 +s 1s ) ∈ 𝐴))
8	7	rspccva 3634	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦 +s 1s ) ∈ 𝐴)
9	8	adantll 713	. 2 ⊢ ((( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 +s 1s ) ∈ 𝐴)
10	2, 4, 5, 9	noseqind 28316	1 ⊢ (( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) → ℕ0s ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∀wral 3067  Vcvv 3488   ⊆ wss 3976   ↦ cmpt 5249   “ cima 5703  (class class class)co 7448  ωcom 7903  reccrdg 8465   No csur 27702   0_s c0s 27885   1_s c1s 27886   +_s cadds 28010  ℕ_0scnn0s 28336

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-2o 8523  df-no 27705  df-slt 27706  df-bday 27707  df-sslt 27844  df-scut 27846  df-0s 27887  df-n0s 28338

This theorem is referenced by:  dfn0s2  28354