Theorem bj-axun2 10706

Description: axun2 4190 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-axun2	|- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )

Distinct variable group:   x, w, y, z

Proof of Theorem bj-axun2

Step	Hyp	Ref	Expression
1		ax-bdel 10612	. . . 4 |- BOUNDED z e. w
2	1	ax-bdex 10610	. . 3 |- BOUNDED E. w e. x z e. w
3		df-rex 2354	. . . 4 |- ( E. w e. x z e. w <-> E. w ( w e. x /\ z e. w ) )
4		exancom 1539	. . . 4 |- ( E. w ( w e. x /\ z e. w ) <-> E. w ( z e. w /\ w e. x ) )
5	3, 4	bitri 182	. . 3 |- ( E. w e. x z e. w <-> E. w ( z e. w /\ w e. x ) )
6	2, 5	bd0 10615	. 2 |- BOUNDED E. w ( z e. w /\ w e. x )
7		ax-un 4188	. 2 |- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
8	6, 7	bdbm1.3ii 10682	1 |- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )

Syntax hints:   /\ wa 102   <-> wb 103   A.wal 1282   E.wex 1421   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-un 4188  ax-bd0 10604  ax-bdex 10610  ax-bdel 10612  ax-bdsep 10675

This theorem depends on definitions:  df-bi 115  df-rex 2354

This theorem is referenced by: (None)