Theorem bj-axempty2 15540

Description: Axiom of the empty set from bounded separation, alternate version to bj-axempty 15539. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4159 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 15538	. 2 |- E. x A. y ( y e. x -> F. )
2		dfnot 1382	. . . 4 |- ( -. y e. x <-> ( y e. x -> F. ) )
3	2	albii 1484	. . 3 |- ( A. y -. y e. x <-> A. y ( y e. x -> F. ) )
4	3	exbii 1619	. 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 1362   F. wfal 1369   E.wex 1506

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 615  ax-in2 616  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548  ax-bd0 15459  ax-bdim 15460  ax-bdn 15463  ax-bdeq 15466  ax-bdsep 15530

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370

This theorem is referenced by: (None)