Theorem dfn0s2 28652

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 28129	. . . . . 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 1832	. . 3 ⊢ 0s ∈ ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}
6		oveq1 7415	. . . . . . . . . 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 1844	. . . . 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 7441	. . . . . 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 28639	. . 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 28649	. . . 4 ⊢ 0s ∈ ℕ0s
21		peano2n0s 28650	. . . . 5 ⊢ (𝑦 ∈ ℕ0s → (𝑦 +s 1s ) ∈ ℕ0s)
22	21	rgen 3078	. . . 4 ⊢ ∀𝑦 ∈ ℕ0s (𝑦 +s 1s ) ∈ ℕ0s
23		n0sex 28637	. . . . 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 3946	1 ⊢ ℕ0s = ∩ {𝑥 ∣ ( 0s ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 +s 1s ) ∈ 𝑥)}

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076   ⊆ wss 3898  ∩ cint 4906  (class class class)co 7408   No csur 27931   0_s c0s 28125   1_s c1s 28126   +_s cadds 28279  ℕ_0scn0s 28632

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  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 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-2o 8455  df-no 27934  df-lts 27935  df-bday 27936  df-slts 28078  df-cuts 28080  df-0s 28127  df-n0s 28634

This theorem is used by: (None)