Theorem dfn0s2 28719

Description: Alternate definition of the set of non-negative surreal integers. (Contributed by Scott Fenton, 17-Mar-2025.)

Assertion

Ref	Expression
dfn0s2	⊢ ℕ0s = ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}

Distinct variable group:   𝑥,𝑦

Proof of Theorem dfn0s2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0no 28196	. . . . . 6 ⊢ 0s ∈ No
2	1	elexi 3472	. . . . 5 ⊢ 0s ∈ V
3	2	elintab 4918	. . . 4 ⊢ ( 0s ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ∀𝑥(( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 0s ∈ 𝑥))
4		simpl 488	. . . 4 ⊢ (( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 0s ∈ 𝑥)
5	3, 4	mpgbir 1831	. . 3 ⊢ 0s ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
6		oveq1 7427	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑦 +s 1s ) = (𝑧 +s 1s ))
7	6	eleq1d 2845	. . . . . . . . 9 ⊢ (𝑦 = 𝑧 → ((𝑦 +s 1s ) ∈ 𝑥 ↔ (𝑧 +s 1s ) ∈ 𝑥))
8	7	rspccv 3573	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥 → (𝑧 ∈ 𝑥 → (𝑧 +s 1s ) ∈ 𝑥))
9	8	adantl 487	. . . . . . 7 ⊢ (( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 ∈ 𝑥 → (𝑧 +s 1s ) ∈ 𝑥))
10	9	a2i 15	. . . . . 6 ⊢ ((( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 𝑧 ∈ 𝑥) → (( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 +s 1s ) ∈ 𝑥))
11	10	alimi 1843	. . . . 5 ⊢ (∀𝑥(( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 𝑧 ∈ 𝑥) → ∀𝑥(( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 +s 1s ) ∈ 𝑥))
12		vex 3454	. . . . . 6 ⊢ 𝑧 ∈ V
13	12	elintab 4918	. . . . 5 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ∀𝑥(( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → 𝑧 ∈ 𝑥))
14		ovex 7453	. . . . . 6 ⊢ (𝑧 +s 1s ) ∈ V
15	14	elintab 4918	. . . . 5 ⊢ ((𝑧 +s 1s ) ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ∀𝑥(( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) → (𝑧 +s 1s ) ∈ 𝑥))
16	11, 13, 15	3imtr4i 295	. . . 4 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} → (𝑧 +s 1s ) ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)})
17	16	rgen 3078	. . 3 ⊢ ∀𝑧 ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} (𝑧 +s 1s ) ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
18		peano5n0s 28706	. . 3 ⊢ (( 0s ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ∧ ∀𝑧 ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} (𝑧 +s 1s ) ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}) → ℕ0s ⊆ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)})
19	5, 17, 18	mp2an 705	. 2 ⊢ ℕ0s ⊆ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
20		0n0s 28716	. . . 4 ⊢ 0s ∈ ℕ0s
21		peano2n0s 28717	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
22	21	rgen 3078	. . . 4 ⊢ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s
23		n0sex 28704	. . . . 5 ⊢ ℕ0s ∈ V
24		eleq2 2849	. . . . . 6 ⊢ (𝑥 = ℕ0s → ( 0s ∈ 𝑥 ↔ 0s ∈ ℕ0s))
25		eleq2 2849	. . . . . . 7 ⊢ (𝑥 = ℕ0s → ((𝑦 +s 1s ) ∈ 𝑥 ↔ (𝑦 +s 1s ) ∈ ℕ0s))
26	25	raleqbi1dv 3329	. . . . . 6 ⊢ (𝑥 = ℕ0s → (∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥 ↔ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s))
27	24, 26	anbi12d 644	. . . . 5 ⊢ (𝑥 = ℕ0s → (( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥) ↔ ( 0s ∈ ℕ0s ∧ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s)))
28	23, 27	elab 3632	. . . 4 ⊢ (ℕ0s ∈ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ↔ ( 0s ∈ ℕ0s ∧ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s))
29	20, 22, 28	mpbir2an 724	. . 3 ⊢ ℕ0s ∈ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
30		intss1 4922	. . 3 ⊢ (ℕ0s ∈ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} → ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ⊆ ℕ0s)
31	29, 30	ax-mp 5	. 2 ⊢ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)} ⊆ ℕ0s
32	19, 31	eqssi 3924	1 ⊢ ℕ0s = ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076   ⊆ wss 3895  ∩ cint 4906  (class class class)co 7420  Nocsur 27998   0_s c0s 28192   1_s c1s 28193   +_s cadds 28346  ℕ_0scn0s 28699

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9643

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-2o 8478  df-no 28001  df-lts 28002  df-bday 28003  df-slts 28145  df-cuts 28147  df-0s 28194  df-n0s 28701

This theorem is used by: (None)