Theorem bj-indind 16648

Description: If 𝐴 is inductive and 𝐵 is "inductive in 𝐴 " (a condition weaker than "inductive"), then (𝐴 ∩ 𝐵) is inductive. (Contributed by BJ, 25-Oct-2020.)

Assertion

Ref	Expression
bj-indind	⊢ ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))) → Ind (𝐴 ∩ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bj-indind

Step	Hyp	Ref	Expression
1		df-bj-ind 16643	. . . 4 ⊢ (Ind 𝐴 ↔ (∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴))
2		id 19	. . . . 5 ⊢ (((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))))
3	2	an4s 592	. . . 4 ⊢ (((∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) ∧ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))))
4	1, 3	sylanb 284	. . 3 ⊢ ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))))
5		elin 3392	. . . . 5 ⊢ (∅ ∈ (𝐴 ∩ 𝐵) ↔ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))
6	5	biimpri 133	. . . 4 ⊢ ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴 ∩ 𝐵))
7		r19.26 2660	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 (suc 𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵)) ↔ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵)))
8	7	biimpri 133	. . . . . . 7 ⊢ ((∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 (suc 𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵)))
9		simpl 109	. . . . . . . . 9 ⊢ ((suc 𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵)) → suc 𝑥 ∈ 𝐴)
10		simpr 110	. . . . . . . . 9 ⊢ ((suc 𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵)) → (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))
11		elin 3392	. . . . . . . . . 10 ⊢ (suc 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (suc 𝑥 ∈ 𝐴 ∧ suc 𝑥 ∈ 𝐵))
12	11	biimpri 133	. . . . . . . . 9 ⊢ ((suc 𝑥 ∈ 𝐴 ∧ suc 𝑥 ∈ 𝐵) → suc 𝑥 ∈ (𝐴 ∩ 𝐵))
13	9, 10, 12	syl6an 1479	. . . . . . . 8 ⊢ ((suc 𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵)) → (𝑥 ∈ 𝐵 → suc 𝑥 ∈ (𝐴 ∩ 𝐵)))
14	13	ralimi 2596	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (suc 𝑥 ∈ 𝐴 ∧ (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ (𝐴 ∩ 𝐵)))
15	8, 14	syl 14	. . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵)) → ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ (𝐴 ∩ 𝐵)))
16		df-ral 2516	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ (𝐴 ∩ 𝐵)) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → suc 𝑥 ∈ (𝐴 ∩ 𝐵))))
17		elin 3392	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∩ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
18		pm3.31 262	. . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → suc 𝑥 ∈ (𝐴 ∩ 𝐵))) → ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵) → suc 𝑥 ∈ (𝐴 ∩ 𝐵)))
19	17, 18	biimtrid 152	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → suc 𝑥 ∈ (𝐴 ∩ 𝐵))) → (𝑥 ∈ (𝐴 ∩ 𝐵) → suc 𝑥 ∈ (𝐴 ∩ 𝐵)))
20	19	alimi 1504	. . . . . . 7 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐵 → suc 𝑥 ∈ (𝐴 ∩ 𝐵))) → ∀𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) → suc 𝑥 ∈ (𝐴 ∩ 𝐵)))
21	16, 20	sylbi 121	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ (𝐴 ∩ 𝐵)) → ∀𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) → suc 𝑥 ∈ (𝐴 ∩ 𝐵)))
22	15, 21	syl 14	. . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵)) → ∀𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) → suc 𝑥 ∈ (𝐴 ∩ 𝐵)))
23		df-ral 2516	. . . . 5 ⊢ (∀𝑥 ∈ (𝐴 ∩ 𝐵)suc 𝑥 ∈ (𝐴 ∩ 𝐵) ↔ ∀𝑥(𝑥 ∈ (𝐴 ∩ 𝐵) → suc 𝑥 ∈ (𝐴 ∩ 𝐵)))
24	22, 23	sylibr 134	. . . 4 ⊢ ((∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵)) → ∀𝑥 ∈ (𝐴 ∩ 𝐵)suc 𝑥 ∈ (𝐴 ∩ 𝐵))
25	6, 24	anim12i 338	. . 3 ⊢ (((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) ∧ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))) → (∅ ∈ (𝐴 ∩ 𝐵) ∧ ∀𝑥 ∈ (𝐴 ∩ 𝐵)suc 𝑥 ∈ (𝐴 ∩ 𝐵)))
26	4, 25	syl 14	. 2 ⊢ ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))) → (∅ ∈ (𝐴 ∩ 𝐵) ∧ ∀𝑥 ∈ (𝐴 ∩ 𝐵)suc 𝑥 ∈ (𝐴 ∩ 𝐵)))
27		df-bj-ind 16643	. 2 ⊢ (Ind (𝐴 ∩ 𝐵) ↔ (∅ ∈ (𝐴 ∩ 𝐵) ∧ ∀𝑥 ∈ (𝐴 ∩ 𝐵)suc 𝑥 ∈ (𝐴 ∩ 𝐵)))
28	26, 27	sylibr 134	1 ⊢ ((Ind 𝐴 ∧ (∅ ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝐵 → suc 𝑥 ∈ 𝐵))) → Ind (𝐴 ∩ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1396   ∈ wcel 2202  ∀wral 2511   ∩ cin 3200  ∅c0 3496  suc csuc 4468  Ind wind 16642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-in 3207  df-bj-ind 16643

This theorem is referenced by:  peano5set  16656