Theorem peano5n0s 28399

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 28394	. . 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 27889	. . 3 ⊢ 0s ∈ No
4	3	a1i 11	. 2 ⊢ (( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) → 0s ∈ No )
5		simpl 486	. 2 ⊢ (( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) → 0s ∈ 𝐴)
6		oveq1 7397	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 +s 1s ) = (𝑦 +s 1s ))
7	6	eleq1d 2846	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 +s 1s ) ∈ 𝐴 ↔ (𝑦 +s 1s ) ∈ 𝐴))
8	7	rspccva 3579	. . 3 ⊢ ((∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦 +s 1s ) ∈ 𝐴)
9	8	adantll 724	. 2 ⊢ ((( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → (𝑦 +s 1s ) ∈ 𝐴)
10	2, 4, 5, 9	noseqind 28372	1 ⊢ (( 0s ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 +s 1s ) ∈ 𝐴) → ℕ0s ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  Vcvv 3453   ⊆ wss 3902   ↦ cmpt 5178   “ cima 5646  (class class class)co 7390  ωcom 7840  reccrdg 8373   No csur 27691   0_s c0s 27885   1_s c1s 27886   +_s cadds 28039  ℕ_0scn0s 28392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4903  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-1o 8430  df-2o 8431  df-no 27694  df-lts 27695  df-bday 27696  df-slts 27838  df-cuts 27840  df-0s 27887  df-n0s 28394

This theorem is referenced by:  dfn0s2  28412