Theorem bdinex1 9354

Description: Bounded version of inex1 3882. (Contributed by BJ, 13-Nov-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bdinex1.bd	|- BOUNDED B
bdinex1.1	|- A e. _V

Assertion

Ref	Expression
bdinex1	|- (A i^i B) e. _V

Proof of Theorem bdinex1

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdinex1.1	. . . 4 |- A e. _V
2		bdinex1.bd	. . . . . 6 |- BOUNDED B
3	2	bdeli 9301	. . . . 5 |- BOUNDED y e. B
4	3	bdzfauscl 9345	. . . 4 |- (A e. _V -> E.xA.y(y e. x <-> (y e. A /\ y e. B)))
5	1, 4	ax-mp 7	. . 3 |- E.xA.y(y e. x <-> (y e. A /\ y e. B))
6		dfcleq 2031	. . . . 5 |- (x = (A i^i B) <-> A.y(y e. x <-> y e. (A i^i B)))
7		elin 3120	. . . . . . 7 |- (y e. (A i^i B) <-> (y e. A /\ y e. B))
8	7	bibi2i 216	. . . . . 6 |- ((y e. x <-> y e. (A i^i B)) <-> (y e. x <-> (y e. A /\ y e. B)))
9	8	albii 1356	. . . . 5 |- (A.y(y e. x <-> y e. (A i^i B)) <-> A.y(y e. x <-> (y e. A /\ y e. B)))
10	6, 9	bitri 173	. . . 4 |- (x = (A i^i B) <-> A.y(y e. x <-> (y e. A /\ y e. B)))
11	10	exbii 1493	. . 3 |- (E.x x = (A i^i B) <-> E.xA.y(y e. x <-> (y e. A /\ y e. B)))
12	5, 11	mpbir 134	. 2 |- E.x x = (A i^i B)
13	12	issetri 2558	1 |- (A i^i B) e. _V

Syntax hints:   /\ wa 97   <-> wb 98  A.wal 1240   = wceq 1242  E.wex 1378   e. wcel 1390   _Vcvv 2551   i^i cin 2910  BOUNDED wbdc 9295

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-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-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  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-v 2553  df-in 2918  df-bdc 9296

This theorem is referenced by:  bdinex2  9355  bdinex1g  9356  bdpeano5  9403