Theorem bdinex1 16286

Description: Bounded version of inex1 4218. (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 16233	. . . . 5 |- BOUNDED y e. B
4	3	bdzfauscl 16277	. . . 4 |- ( A e. _V -> E. x A. y ( y e. x <-> ( y e. A /\ y e. B ) ) )
5	1, 4	ax-mp 5	. . 3 |- E. x A. y ( y e. x <-> ( y e. A /\ y e. B ) )
6		dfcleq 2223	. . . . 5 |- ( x = ( A i^i B ) <-> A. y ( y e. x <-> y e. ( A i^i B ) ) )
7		elin 3387	. . . . . . 7 |- ( y e. ( A i^i B ) <-> ( y e. A /\ y e. B ) )
8	7	bibi2i 227	. . . . . 6 |- ( ( y e. x <-> y e. ( A i^i B ) ) <-> ( y e. x <-> ( y e. A /\ y e. B ) ) )
9	8	albii 1516	. . . . 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 184	. . . 4 |- ( x = ( A i^i B ) <-> A. y ( y e. x <-> ( y e. A /\ y e. B ) ) )
11	10	exbii 1651	. . 3 |- ( E. x x = ( A i^i B ) <-> E. x A. y ( y e. x <-> ( y e. A /\ y e. B ) ) )
12	5, 11	mpbir 146	. 2 |- E. x x = ( A i^i B )
13	12	issetri 2809	1 |- ( A i^i B ) e. _V

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1393   = wceq 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   i^i cin 3196  BOUNDED wbdc 16227

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bdsep 16271

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-in 3203  df-bdc 16228

This theorem is referenced by:  bdinex2  16287  bdinex1g  16288  bdpeano5  16330