Theorem bj-axempty2 14807

Description: Axiom of the empty set from bounded separation, alternate version to bj-axempty 14806. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4131 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 14805	. 2 |- E. x A. y ( y e. x -> F. )
2		dfnot 1371	. . . 4 |- ( -. y e. x <-> ( y e. x -> F. ) )
3	2	albii 1470	. . 3 |- ( A. y -. y e. x <-> A. y ( y e. x -> F. ) )
4	3	exbii 1605	. 2 |- ( E. x A. y -. y e. x <-> E. x A. y ( y e. x -> F. ) )
5	1, 4	mpbir 146	1 |- E. x A. y -. y e. x

Syntax hints:   -. wn 3   -> wi 4   A.wal 1351   F. wfal 1358   E.wex 1492

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-in1 614  ax-in2 615  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534  ax-bd0 14726  ax-bdim 14727  ax-bdn 14730  ax-bdeq 14733  ax-bdsep 14797

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-fal 1359

This theorem is referenced by: (None)