ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elsuc2g Unicode version

Theorem elsuc2g 4285
Description: Variant of membership in a successor, requiring that  B rather than  A be a set. (Contributed by NM, 28-Oct-2003.)
Assertion
Ref Expression
elsuc2g  |-  ( B  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )

Proof of Theorem elsuc2g
StepHypRef Expression
1 df-suc 4251 . . 3  |-  suc  B  =  ( B  u.  { B } )
21eleq2i 2179 . 2  |-  ( A  e.  suc  B  <->  A  e.  ( B  u.  { B } ) )
3 elun 3181 . . 3  |-  ( A  e.  ( B  u.  { B } )  <->  ( A  e.  B  \/  A  e.  { B } ) )
4 elsn2g 3522 . . . 4  |-  ( B  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
54orbi2d 762 . . 3  |-  ( B  e.  V  ->  (
( A  e.  B  \/  A  e.  { B } )  <->  ( A  e.  B  \/  A  =  B ) ) )
63, 5syl5bb 191 . 2  |-  ( B  e.  V  ->  ( A  e.  ( B  u.  { B } )  <-> 
( A  e.  B  \/  A  =  B
) ) )
72, 6syl5bb 191 1  |-  ( B  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 680    = wceq 1312    e. wcel 1461    u. cun 3033   {csn 3491   suc csuc 4245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-un 3039  df-sn 3497  df-suc 4251
This theorem is referenced by:  elsuc2  4287  nntri3or  6341  frec2uzltd  10063
  Copyright terms: Public domain W3C validator