Theorem bj-axempty2 10014

Description: Axiom of the empty set from bounded separation, alternate version to bj-axempty 10013. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 instead. (New usage is discouraged.)

Assertion

Ref	Expression
bj-axempty2	|- E. x A. y -. y e. x

Distinct variable group:   x, y

Proof of Theorem bj-axempty2

Step	Hyp	Ref	Expression
1		bj-axemptylem 10012	. 2 |- E. x A. y ( y e. x -> F. )
2		dfnot 1262	. . . 4 |- ( -. y e. x <-> ( y e. x -> F. ) )
3	2	albii 1359	. . 3 |- ( A. y -. y e. x <-> A. y ( y e. x -> F. ) )
4	3	exbii 1496	. 2 |- ( E. x A. y -. y e. x <-> E. x A. y ( y e. x -> F. ) )
5	1, 4	mpbir 134	1 |- E. x A. y -. y e. x

Syntax hints:   -. wn 3   -> wi 4   A.wal 1241   F. wfal 1248   E.wex 1381

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-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-bd0 9933  ax-bdim 9934  ax-bdn 9937  ax-bdeq 9940  ax-bdsep 10004

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249

This theorem is referenced by: (None)