Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >   Mathboxes  >  bj-axempty Unicode version

Theorem bj-axempty 11428
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 3956. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3957 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 11427 . 2  |-  E. x A. y ( y  e.  x  -> F.  )
2 df-ral 2364 . . 3  |-  ( A. y  e.  x F.  <->  A. y ( y  e.  x  -> F.  )
)
32exbii 1541 . 2  |-  ( E. x A. y  e.  x F.  <->  E. x A. y ( y  e.  x  -> F.  )
)
41, 3mpbir 144 1  |-  E. x A. y  e.  x F.
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1287   F. wfal 1294   E.wex 1426   A.wral 2359
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472  ax-bd0 11348  ax-bdim 11349  ax-bdn 11352  ax-bdeq 11355  ax-bdsep 11419
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-ral 2364
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator