Theorem bj-axempty 14916

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 4140. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4141 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 14915	. 2 |- E. x A. y ( y e. x -> F. )
2		df-ral 2470	. . 3 |- ( A. y e. x F. <-> A. y ( y e. x -> F. ) )
3	2	exbii 1615	. 2 |- ( E. x A. y e. x F. <-> E. x A. y ( y e. x -> F. ) )
4	1, 3	mpbir 146	1 |- E. x A. y e. x F.

Syntax hints:   -> wi 4   A.wal 1361   F. wfal 1368   E.wex 1502   A.wral 2465

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 1457  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-ial 1544  ax-bd0 14836  ax-bdim 14837  ax-bdn 14840  ax-bdeq 14843  ax-bdsep 14907

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-ral 2470

This theorem is referenced by: (None)