MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nrexrmo Unicode version

Theorem nrexrmo 2757
Description: Nonexistence implies restricted "at most one". (Contributed by NM, 17-Jun-2017.)
Assertion
Ref Expression
nrexrmo  |-  ( -. 
E. x  e.  A  ph 
->  E* x  e.  A ph )

Proof of Theorem nrexrmo
StepHypRef Expression
1 pm2.21 100 . 2  |-  ( -. 
E. x  e.  A  ph 
->  ( E. x  e.  A  ph  ->  E! x  e.  A  ph )
)
2 rmo5 2756 . 2  |-  ( E* x  e.  A ph  <->  ( E. x  e.  A  ph 
->  E! x  e.  A  ph ) )
31, 2sylibr 203 1  |-  ( -. 
E. x  e.  A  ph 
->  E* x  e.  A ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wrex 2544   E!wreu 2545   E*wrmo 2546
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-mo 2148  df-rex 2549  df-reu 2550  df-rmo 2551
  Copyright terms: Public domain W3C validator