Theorem bj-axempty 16214

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 4208. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4209 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 16213	. 2 |- E. x A. y ( y e. x -> F. )
2		df-ral 2513	. . 3 |- ( A. y e. x F. <-> A. y ( y e. x -> F. ) )
3	2	exbii 1651	. 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 1393   F. wfal 1400   E.wex 1538   A.wral 2508

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 617  ax-in2 618  ax-5 1493  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-4 1556  ax-ial 1580  ax-bd0 16134  ax-bdim 16135  ax-bdn 16138  ax-bdeq 16141  ax-bdsep 16205

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-ral 2513

This theorem is referenced by: (None)