Theorem bj-indint 15169

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 15165	. . . . 5 ⊢ (Ind 𝑥 ↔ (∅ ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 suc 𝑦 ∈ 𝑥))
2	1	simplbi 274	. . . 4 ⊢ (Ind 𝑥 → ∅ ∈ 𝑥)
3	2	rgenw 2545	. . 3 ⊢ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥)
4		0ex 4148	. . . 4 ⊢ ∅ ∈ V
5	4	elintrab 3874	. . 3 ⊢ (∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → ∅ ∈ 𝑥))
6	3, 5	mpbir 146	. 2 ⊢ ∅ ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}
7		bj-indsuc 15166	. . . . . 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 2553	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥) → ∀𝑥 ∈ 𝐴 (Ind 𝑥 → suc 𝑦 ∈ 𝑥))
10		vex 2755	. . . . 5 ⊢ 𝑦 ∈ V
11	10	elintrab 3874	. . . 4 ⊢ (𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥} ↔ ∀𝑥 ∈ 𝐴 (Ind 𝑥 → 𝑦 ∈ 𝑥))
12	10	bj-sucex 15161	. . . . 5 ⊢ suc 𝑦 ∈ V
13	12	elintrab 3874	. . . 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 2543	. 2 ⊢ ∀𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}suc 𝑦 ∈ ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}
16		df-bj-ind 15165	. 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 944	1 ⊢ Ind ∩ {𝑥 ∈ 𝐴 ∣ Ind 𝑥}

Syntax hints:   → wi 4   ∈ wcel 2160  ∀wral 2468  {crab 2472  ∅c0 3437  ∩ cint 3862  suc csuc 4386  Ind wind 15164

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-nul 4147  ax-pr 4230  ax-un 4454  ax-bd0 15051  ax-bdor 15054  ax-bdex 15057  ax-bdeq 15058  ax-bdel 15059  ax-bdsb 15060  ax-bdsep 15122

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-nul 3438  df-sn 3616  df-pr 3617  df-uni 3828  df-int 3863  df-suc 4392  df-bdc 15079  df-bj-ind 15165

This theorem is referenced by:  bj-omind  15172