Theorem bj-axempty2 13092

Description: Axiom of the empty set from bounded separation, alternate version to bj-axempty 13091. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4054 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 13090	. 2 |- E. x A. y ( y e. x -> F. )
2		dfnot 1349	. . . 4 |- ( -. y e. x <-> ( y e. x -> F. ) )
3	2	albii 1446	. . 3 |- ( A. y -. y e. x <-> A. y ( y e. x -> F. ) )
4	3	exbii 1584	. 2 |- ( E. x A. y -. y e. x <-> E. x A. y ( y e. x -> F. ) )
5	1, 4	mpbir 145	1 |- E. x A. y -. y e. x

Syntax hints:   -. wn 3   -> wi 4   A.wal 1329   F. wfal 1336   E.wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514  ax-bd0 13011  ax-bdim 13012  ax-bdn 13015  ax-bdeq 13018  ax-bdsep 13082

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337

This theorem is referenced by: (None)