Theorem bj-axun2 13797

Description: axun2 4413 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 13703	. . . 4 |- BOUNDED z e. w
2	1	ax-bdex 13701	. . 3 |- BOUNDED E. w e. x z e. w
3		df-rex 2450	. . . 4 |- ( E. w e. x z e. w <-> E. w ( w e. x /\ z e. w ) )
4		exancom 1596	. . . 4 |- ( E. w ( w e. x /\ z e. w ) <-> E. w ( z e. w /\ w e. x ) )
5	3, 4	bitri 183	. . 3 |- ( E. w e. x z e. w <-> E. w ( z e. w /\ w e. x ) )
6	2, 5	bd0 13706	. 2 |- BOUNDED E. w ( z e. w /\ w e. x )
7		ax-un 4411	. 2 |- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
8	6, 7	bdbm1.3ii 13773	1 |- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )

Syntax hints:   /\ wa 103   <-> wb 104   A.wal 1341   E.wex 1480   E.wrex 2445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-14 2139  ax-un 4411  ax-bd0 13695  ax-bdex 13701  ax-bdel 13703  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-rex 2450

This theorem is referenced by: (None)