Theorem bdinex1 14736

Description: Bounded version of inex1 4139. (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 14683	. . . . 5 |- BOUNDED y e. B
4	3	bdzfauscl 14727	. . . 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 2171	. . . . 5 |- ( x = ( A i^i B ) <-> A. y ( y e. x <-> y e. ( A i^i B ) ) )
7		elin 3320	. . . . . . 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 1470	. . . . 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 1605	. . 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 2748	1 |- ( A i^i B ) e. _V

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1351   = wceq 1353   E.wex 1492   e. wcel 2148   _Vcvv 2739   i^i cin 3130  BOUNDED wbdc 14677

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bdsep 14721

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-bdc 14678

This theorem is referenced by:  bdinex2  14737  bdinex1g  14738  bdpeano5  14780