Theorem peano5set 9328

Description: Version of peano5 4264 when 𝜔 ∩ A is assumed to be a set, allowing a proof from the core axioms of CZF. (Contributed by BJ, 19-Nov-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
peano5set	⊢ ((𝜔 ∩ A) ∈ 𝑉 → ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → 𝜔 ⊆ A))

Distinct variable group:   x,A

Allowed substitution hint:   𝑉(x)

Proof of Theorem peano5set

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfom3 4258	. . . 4 ⊢ 𝜔 = ∩ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)}
2		peano1 4260	. . . . . . . . . 10 ⊢ ∅ ∈ 𝜔
3		elin 3120	. . . . . . . . . 10 ⊢ (∅ ∈ (𝜔 ∩ A) ↔ (∅ ∈ 𝜔 ∧ ∅ ∈ A))
4	2, 3	mpbiran 846	. . . . . . . . 9 ⊢ (∅ ∈ (𝜔 ∩ A) ↔ ∅ ∈ A)
5	4	biimpri 124	. . . . . . . 8 ⊢ (∅ ∈ A → ∅ ∈ (𝜔 ∩ A))
6		bj-peano2 9327	. . . . . . . . . . . . . 14 ⊢ (x ∈ 𝜔 → suc x ∈ 𝜔)
7	6	adantr 261	. . . . . . . . . . . . 13 ⊢ ((x ∈ 𝜔 ∧ x ∈ A) → suc x ∈ 𝜔)
8	7	a1i 9	. . . . . . . . . . . 12 ⊢ ((x ∈ 𝜔 → (x ∈ A → suc x ∈ A)) → ((x ∈ 𝜔 ∧ x ∈ A) → suc x ∈ 𝜔))
9		pm3.31 249	. . . . . . . . . . . 12 ⊢ ((x ∈ 𝜔 → (x ∈ A → suc x ∈ A)) → ((x ∈ 𝜔 ∧ x ∈ A) → suc x ∈ A))
10	8, 9	jcad 291	. . . . . . . . . . 11 ⊢ ((x ∈ 𝜔 → (x ∈ A → suc x ∈ A)) → ((x ∈ 𝜔 ∧ x ∈ A) → (suc x ∈ 𝜔 ∧ suc x ∈ A)))
11	10	alimi 1341	. . . . . . . . . 10 ⊢ (∀x(x ∈ 𝜔 → (x ∈ A → suc x ∈ A)) → ∀x((x ∈ 𝜔 ∧ x ∈ A) → (suc x ∈ 𝜔 ∧ suc x ∈ A)))
12		df-ral 2305	. . . . . . . . . 10 ⊢ (∀x ∈ 𝜔 (x ∈ A → suc x ∈ A) ↔ ∀x(x ∈ 𝜔 → (x ∈ A → suc x ∈ A)))
13		elin 3120	. . . . . . . . . . . 12 ⊢ (x ∈ (𝜔 ∩ A) ↔ (x ∈ 𝜔 ∧ x ∈ A))
14		elin 3120	. . . . . . . . . . . 12 ⊢ (suc x ∈ (𝜔 ∩ A) ↔ (suc x ∈ 𝜔 ∧ suc x ∈ A))
15	13, 14	imbi12i 228	. . . . . . . . . . 11 ⊢ ((x ∈ (𝜔 ∩ A) → suc x ∈ (𝜔 ∩ A)) ↔ ((x ∈ 𝜔 ∧ x ∈ A) → (suc x ∈ 𝜔 ∧ suc x ∈ A)))
16	15	albii 1356	. . . . . . . . . 10 ⊢ (∀x(x ∈ (𝜔 ∩ A) → suc x ∈ (𝜔 ∩ A)) ↔ ∀x((x ∈ 𝜔 ∧ x ∈ A) → (suc x ∈ 𝜔 ∧ suc x ∈ A)))
17	11, 12, 16	3imtr4i 190	. . . . . . . . 9 ⊢ (∀x ∈ 𝜔 (x ∈ A → suc x ∈ A) → ∀x(x ∈ (𝜔 ∩ A) → suc x ∈ (𝜔 ∩ A)))
18		df-ral 2305	. . . . . . . . 9 ⊢ (∀x ∈ (𝜔 ∩ A)suc x ∈ (𝜔 ∩ A) ↔ ∀x(x ∈ (𝜔 ∩ A) → suc x ∈ (𝜔 ∩ A)))
19	17, 18	sylibr 137	. . . . . . . 8 ⊢ (∀x ∈ 𝜔 (x ∈ A → suc x ∈ A) → ∀x ∈ (𝜔 ∩ A)suc x ∈ (𝜔 ∩ A))
20	5, 19	anim12i 321	. . . . . . 7 ⊢ ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → (∅ ∈ (𝜔 ∩ A) ∧ ∀x ∈ (𝜔 ∩ A)suc x ∈ (𝜔 ∩ A)))
21		eleq2 2098	. . . . . . . . 9 ⊢ (y = (𝜔 ∩ A) → (∅ ∈ y ↔ ∅ ∈ (𝜔 ∩ A)))
22		eleq2 2098	. . . . . . . . . 10 ⊢ (y = (𝜔 ∩ A) → (suc x ∈ y ↔ suc x ∈ (𝜔 ∩ A)))
23	22	raleqbi1dv 2507	. . . . . . . . 9 ⊢ (y = (𝜔 ∩ A) → (∀x ∈ y suc x ∈ y ↔ ∀x ∈ (𝜔 ∩ A)suc x ∈ (𝜔 ∩ A)))
24	21, 23	anbi12d 442	. . . . . . . 8 ⊢ (y = (𝜔 ∩ A) → ((∅ ∈ y ∧ ∀x ∈ y suc x ∈ y) ↔ (∅ ∈ (𝜔 ∩ A) ∧ ∀x ∈ (𝜔 ∩ A)suc x ∈ (𝜔 ∩ A))))
25	24	elabg 2682	. . . . . . 7 ⊢ ((𝜔 ∩ A) ∈ 𝑉 → ((𝜔 ∩ A) ∈ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)} ↔ (∅ ∈ (𝜔 ∩ A) ∧ ∀x ∈ (𝜔 ∩ A)suc x ∈ (𝜔 ∩ A))))
26	20, 25	syl5ibr 145	. . . . . 6 ⊢ ((𝜔 ∩ A) ∈ 𝑉 → ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → (𝜔 ∩ A) ∈ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)}))
27	26	imp 115	. . . . 5 ⊢ (((𝜔 ∩ A) ∈ 𝑉 ∧ (∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A))) → (𝜔 ∩ A) ∈ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)})
28		intss1 3621	. . . . 5 ⊢ ((𝜔 ∩ A) ∈ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)} → ∩ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)} ⊆ (𝜔 ∩ A))
29	27, 28	syl 14	. . . 4 ⊢ (((𝜔 ∩ A) ∈ 𝑉 ∧ (∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A))) → ∩ {y ∣ (∅ ∈ y ∧ ∀x ∈ y suc x ∈ y)} ⊆ (𝜔 ∩ A))
30	1, 29	syl5eqss 2983	. . 3 ⊢ (((𝜔 ∩ A) ∈ 𝑉 ∧ (∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A))) → 𝜔 ⊆ (𝜔 ∩ A))
31		inss2 3152	. . 3 ⊢ (𝜔 ∩ A) ⊆ A
32	30, 31	syl6ss 2951	. 2 ⊢ (((𝜔 ∩ A) ∈ 𝑉 ∧ (∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A))) → 𝜔 ⊆ A)
33	32	ex 108	1 ⊢ ((𝜔 ∩ A) ∈ 𝑉 → ((∅ ∈ A ∧ ∀x ∈ 𝜔 (x ∈ A → suc x ∈ A)) → 𝜔 ⊆ A))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1240   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300   ∩ cin 2910   ⊆ wss 2911  ∅c0 3218  ∩ cint 3606  suc csuc 4068  𝜔com 4256

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-nul 3874  ax-pr 3935  ax-un 4136  ax-bd0 9202  ax-bdor 9205  ax-bdex 9208  ax-bdeq 9209  ax-bdel 9210  ax-bdsb 9211  ax-bdsep 9273

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-iom 4257  df-bdc 9230  df-bj-ind 9316

This theorem is referenced by:  bdpeano5  9331  speano5  9332