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 e. 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 <-> ( (/) e. x /\ A.y e. x suc y e. x))
2	1	simplbi 259	. . . 4 |- (Ind x -> (/) e. x)
3	2	rgenw 2370	. . 3 |- A.x e. A (Ind x -> (/) e. x)
4		0ex 3875	. . . 4 |- (/) e. _V
5	4	elintrab 3618	. . 3 |- ( (/) e. |^| {x e. A | Ind x } <-> A.x e. A (Ind x -> (/) e. x))
6	3, 5	mpbir 134	. 2 |- (/) e. |^| {x e. A | Ind x }
7		bj-indsuc 9387	. . . . . 6 |- (Ind x -> (y e. x -> suc y e. x))
8	7	a2i 11	. . . . 5 |- ((Ind x -> y e. x) -> (Ind x -> suc y e. x))
9	8	ralimi 2378	. . . 4 |- (A.x e. A (Ind x -> y e. x) -> A.x e. A (Ind x -> suc y e. x))
10		vex 2554	. . . . 5 |- y e. _V
11	10	elintrab 3618	. . . 4 |- (y e. |^| {x e. A | Ind x } <-> A.x e. A (Ind x -> y e. x))
12	10	bj-sucex 9378	. . . . 5 |- suc y e. _V
13	12	elintrab 3618	. . . 4 |- ( suc y e. |^| {x e. A | Ind x } <-> A.x e. A (Ind x -> suc y e. x))
14	9, 11, 13	3imtr4i 190	. . 3 |- (y e. |^| {x e. A | Ind x } -> suc y e. |^| {x e. A | Ind x })
15	14	rgen 2368	. 2 |- A.y e. |^| {x e. A | Ind x } suc y e. |^| {x e. A | Ind x }
16		df-bj-ind 9386	. 2 |- (Ind |^| {x e. A | Ind x } <-> ( (/) e. |^| {x e. A | Ind x } /\ A.y e. |^| {x e. A | Ind x } suc y e. |^| {x e. A | Ind x }))
17	6, 15, 16	mpbir2an 848	1 |- Ind |^| {x e. A | Ind x }

Syntax hints:   -> wi 4   e. wcel 1390  A.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