Theorem bj-axun2 15561

Description: axun2 4470 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 15467	. . . 4 |- BOUNDED z e. w
2	1	ax-bdex 15465	. . 3 |- BOUNDED E. w e. x z e. w
3		df-rex 2481	. . . 4 |- ( E. w e. x z e. w <-> E. w ( w e. x /\ z e. w ) )
4		exancom 1622	. . . 4 |- ( E. w ( w e. x /\ z e. w ) <-> E. w ( z e. w /\ w e. x ) )
5	3, 4	bitri 184	. . 3 |- ( E. w e. x z e. w <-> E. w ( z e. w /\ w e. x ) )
6	2, 5	bd0 15470	. 2 |- BOUNDED E. w ( z e. w /\ w e. x )
7		ax-un 4468	. 2 |- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
8	6, 7	bdbm1.3ii 15537	1 |- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )

Syntax hints:   /\ wa 104   <-> wb 105   A.wal 1362   E.wex 1506   E.wrex 2476

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-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-14 2170  ax-un 4468  ax-bd0 15459  ax-bdex 15465  ax-bdel 15467  ax-bdsep 15530

This theorem depends on definitions:  df-bi 117  df-rex 2481

This theorem is referenced by: (None)