Theorem bj-axempty 13928

Description: Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a nonempty universe. See axnul 4114. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4115 instead. (New usage is discouraged.)

Assertion

Ref	Expression
bj-axempty	|- E. x A. y e. x F.

Distinct variable group:   x, y

Proof of Theorem bj-axempty

Step	Hyp	Ref	Expression
1		bj-axemptylem 13927	. 2 |- E. x A. y ( y e. x -> F. )
2		df-ral 2453	. . 3 |- ( A. y e. x F. <-> A. y ( y e. x -> F. ) )
3	2	exbii 1598	. 2 |- ( E. x A. y e. x F. <-> E. x A. y ( y e. x -> F. ) )
4	1, 3	mpbir 145	1 |- E. x A. y e. x F.

Syntax hints:   -> wi 4   A.wal 1346   F. wfal 1353   E.wex 1485   A.wral 2448

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 609  ax-in2 610  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527  ax-bd0 13848  ax-bdim 13849  ax-bdn 13852  ax-bdeq 13855  ax-bdsep 13919

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-ral 2453

This theorem is referenced by: (None)