Theorem bdinex1 7122

Description: Bounded version of inex1 3865. (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 7073	. . . . 5 |- BOUNDED y e. B
4	3	bdzfauscl 7116	. . . 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 2016	. . . . 5 |- (x = (A i^i B) <-> A.y(y e. x <-> y e. (A i^i B)))
7		elin 3103	. . . . . . 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 1339	. . . . 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 1478	. . 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 2542	1 |- (A i^i B) e. _V

Syntax hints:   /\ wa 97   <-> wb 98  A.wal 1226   = wceq 1228  E.wex 1362   e. wcel 1374   _Vcvv 2535   i^i cin 2893  BOUNDED wbdc 7067

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-bdsep 7111

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-bdc 7068

This theorem is referenced by:  bdinex2  7123  bdinex1g  7124  bdpeano5  7165