Theorem bj-indint 9390

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 9386	. . . . 5 ⊢ (Ind x ↔ (∅ ∈ x ∧ ∀y ∈ x suc y ∈ x))
2	1	simplbi 259	. . . 4 ⊢ (Ind x → ∅ ∈ x)
3	2	rgenw 2370	. . 3 ⊢ ∀x ∈ A (Ind x → ∅ ∈ x)
4		0ex 3875	. . . 4 ⊢ ∅ ∈ V
5	4	elintrab 3618	. . 3 ⊢ (∅ ∈ ∩ {x ∈ A ∣ Ind x} ↔ ∀x ∈ A (Ind x → ∅ ∈ x))
6	3, 5	mpbir 134	. 2 ⊢ ∅ ∈ ∩ {x ∈ A ∣ Ind x}
7		bj-indsuc 9387	. . . . . 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 2378	. . . 4 ⊢ (∀x ∈ A (Ind x → y ∈ x) → ∀x ∈ A (Ind x → suc y ∈ x))
10		vex 2554	. . . . 5 ⊢ y ∈ V
11	10	elintrab 3618	. . . 4 ⊢ (y ∈ ∩ {x ∈ A ∣ Ind x} ↔ ∀x ∈ A (Ind x → y ∈ x))
12	10	bj-sucex 9378	. . . . 5 ⊢ suc y ∈ V
13	12	elintrab 3618	. . . 4 ⊢ (suc y ∈ ∩ {x ∈ A ∣ Ind x} ↔ ∀x ∈ A (Ind x → suc y ∈ x))
14	9, 11, 13	3imtr4i 190	. . 3 ⊢ (y ∈ ∩ {x ∈ A ∣ Ind x} → suc y ∈ ∩ {x ∈ A ∣ Ind x})
15	14	rgen 2368	. 2 ⊢ ∀y ∈ ∩ {x ∈ A ∣ Ind x}suc y ∈ ∩ {x ∈ A ∣ Ind x}
16		df-bj-ind 9386	. 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 848	1 ⊢ Ind ∩ {x ∈ A ∣ Ind x}

Syntax hints:   → wi 4   ∈ wcel 1390  ∀wral 2300  {crab 2304  ∅c0 3218  ∩ cint 3606  suc csuc 4068  Ind wind 9385

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-bndl 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 9268  ax-bdor 9271  ax-bdex 9274  ax-bdeq 9275  ax-bdel 9276  ax-bdsb 9277  ax-bdsep 9339

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-nul 3219  df-sn 3373  df-pr 3374  df-uni 3572  df-int 3607  df-suc 4074  df-bdc 9296  df-bj-ind 9386

This theorem is referenced by:  bj-omind  9393