Theorem bj-axun2 10035

Description: axun2 4172 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 9941	. . . 4 |- BOUNDED z e. w
2	1	ax-bdex 9939	. . 3 |- BOUNDED E. w e. x z e. w
3		df-rex 2312	. . . 4 |- ( E. w e. x z e. w <-> E. w ( w e. x /\ z e. w ) )
4		exancom 1499	. . . 4 |- ( E. w ( w e. x /\ z e. w ) <-> E. w ( z e. w /\ w e. x ) )
5	3, 4	bitri 173	. . 3 |- ( E. w e. x z e. w <-> E. w ( z e. w /\ w e. x ) )
6	2, 5	bd0 9944	. 2 |- BOUNDED E. w ( z e. w /\ w e. x )
7		ax-un 4170	. 2 |- E. y A. z ( E. w ( z e. w /\ w e. x ) -> z e. y )
8	6, 7	bdbm1.3ii 10011	1 |- E. y A. z ( z e. y <-> E. w ( z e. w /\ w e. x ) )

Syntax hints:   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381   E.wrex 2307

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-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-un 4170  ax-bd0 9933  ax-bdex 9939  ax-bdel 9941  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-rex 2312

This theorem is referenced by: (None)