Theorem bdinex1 16172

Description: Bounded version of inex1 4197. (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 16119	. . . . 5 |- BOUNDED y e. B
4	3	bdzfauscl 16163	. . . 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 2203	. . . . 5 |- ( x = ( A i^i B ) <-> A. y ( y e. x <-> y e. ( A i^i B ) ) )
7		elin 3367	. . . . . . 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 1496	. . . . 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 1631	. . 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 2789	1 |- ( A i^i B ) e. _V

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1373   = wceq 1375   E.wex 1518   e. wcel 2180   _Vcvv 2779   i^i cin 3176  BOUNDED wbdc 16113

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191  ax-bdsep 16157

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-v 2781  df-in 3183  df-bdc 16114

This theorem is referenced by:  bdinex2  16173  bdinex1g  16174  bdpeano5  16216