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Theorem bj-axempty 10013
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 non-empty universe. See axnul 3882. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3883 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
StepHypRef Expression
1 bj-axemptylem 10012 . 2  |-  E. x A. y ( y  e.  x  -> F.  )
2 df-ral 2311 . . 3  |-  ( A. y  e.  x F.  <->  A. y ( y  e.  x  -> F.  )
)
32exbii 1496 . 2  |-  ( E. x A. y  e.  x F.  <->  E. x A. y ( y  e.  x  -> F.  )
)
41, 3mpbir 134 1  |-  E. x A. y  e.  x F.
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1241   F. wfal 1248   E.wex 1381   A.wral 2306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-bd0 9933  ax-bdim 9934  ax-bdn 9937  ax-bdeq 9940  ax-bdsep 10004
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-ral 2311
This theorem is referenced by: (None)
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