Theorem bj-axempty 13775

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 4107. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4108 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 13774	. 2 |- E. x A. y ( y e. x -> F. )
2		df-ral 2449	. . 3 |- ( A. y e. x F. <-> A. y ( y e. x -> F. ) )
3	2	exbii 1593	. 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 1341   F. wfal 1348   E.wex 1480   A.wral 2444

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 604  ax-in2 605  ax-5 1435  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-4 1498  ax-ial 1522  ax-bd0 13695  ax-bdim 13696  ax-bdn 13699  ax-bdeq 13702  ax-bdsep 13766

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-ral 2449

This theorem is referenced by: (None)