MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elsuc2g Unicode version

Theorem elsuc2g 4617
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 4555 . . 3  |-  suc  B  =  ( B  u.  { B } )
21eleq2i 2476 . 2  |-  ( A  e.  suc  B  <->  A  e.  ( B  u.  { B } ) )
3 elun 3456 . . 3  |-  ( A  e.  ( B  u.  { B } )  <->  ( A  e.  B  \/  A  e.  { B } ) )
4 elsnc2g 3810 . . . 4  |-  ( B  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
54orbi2d 683 . . 3  |-  ( B  e.  V  ->  (
( A  e.  B  \/  A  e.  { B } )  <->  ( A  e.  B  \/  A  =  B ) ) )
63, 5syl5bb 249 . 2  |-  ( B  e.  V  ->  ( A  e.  ( B  u.  { B } )  <-> 
( A  e.  B  \/  A  =  B
) ) )
72, 6syl5bb 249 1  |-  ( B  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    = wceq 1649    e. wcel 1721    u. cun 3286   {csn 3782   suc csuc 4551
This theorem is referenced by:  elsuc2  4619  om2uzlti  11253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926  df-un 3293  df-sn 3788  df-suc 4555
  Copyright terms: Public domain W3C validator