Theorem bj-indint 16630

Description: The property of being an inductive class is closed under intersections. (Contributed by BJ, 30-Nov-2019.)

Assertion

Ref	Expression
bj-indint	⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-indint

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bj-ind 16626	. . . . 5 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥))
2	1	simplbi 274	. . . 4 ⊢ (Ind 𝑥 → ∅ ∈ 𝑥)
3	2	rgenw 2588	. . 3 ⊢ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥)
4		0ex 4221	. . . 4 ⊢ ∅ ∈ V
5	4	elintrab 3945	. . 3 ⊢ (∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥))
6	3, 5	mpbir 146	. 2 ⊢ ∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}
7		bj-indsuc 16627	. . . . . 6 ⊢ (Ind 𝑥 → (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥))
8	7	a2i 11	. . . . 5 ⊢ ((Ind 𝑥 → 𝑦 ∈ 𝑥) → (Ind 𝑥 → suc 𝑦 ∈ 𝑥))
9	8	ralimi 2596	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥) → ∀𝑥 ∈ 𝐴 (Ind 𝑥 → suc 𝑦 ∈ 𝑥))
10		vex 2806	. . . . 5 ⊢ 𝑦 ∈ V
11	10	elintrab 3945	. . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥))
12	10	bj-sucex 16622	. . . . 5 ⊢ suc 𝑦 ∈ V
13	12	elintrab 3945	. . . 4 ⊢ (suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → suc 𝑦 ∈ 𝑥))
14	9, 11, 13	3imtr4i 201	. . 3 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} → suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥})
15	14	rgen 2586	. 2 ⊢ ∀𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}
16		df-bj-ind 16626	. 2 ⊢ (Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ (∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ∧ ∀𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}))
17	6, 15, 16	mpbir2an 951	1 ⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}

Syntax hints:   → wi 4   ∈ wcel 2202  ∀wral 2511  {crab 2515  ∅c0 3496  ∩ cint 3933  suc csuc 4468  Ind wind 16625

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-in1 619  ax-in2 620  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-13 2204  ax-14 2205  ax-ext 2213  ax-nul 4220  ax-pr 4305  ax-un 4536  ax-bd0 16512  ax-bdor 16515  ax-bdex 16518  ax-bdeq 16519  ax-bdel 16520  ax-bdsb 16521  ax-bdsep 16583

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-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-nul 3497  df-sn 3679  df-pr 3680  df-uni 3899  df-int 3934  df-suc 4474  df-bdc 16540  df-bj-ind 16626

This theorem is referenced by:  bj-omind  16633