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Theorem elsuci 4168
Description: Membership in a successor. This one-way implication does not require that either  A or  B be sets. (Contributed by NM, 6-Jun-1994.)
Assertion
Ref Expression
elsuci  |-  ( A  e.  suc  B  -> 
( A  e.  B  \/  A  =  B
) )

Proof of Theorem elsuci
StepHypRef Expression
1 df-suc 4136 . . . 4  |-  suc  B  =  ( B  u.  { B } )
21eleq2i 2120 . . 3  |-  ( A  e.  suc  B  <->  A  e.  ( B  u.  { B } ) )
3 elun 3112 . . 3  |-  ( A  e.  ( B  u.  { B } )  <->  ( A  e.  B  \/  A  e.  { B } ) )
42, 3bitri 177 . 2  |-  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  e.  { B } ) )
5 elsni 3421 . . 3  |-  ( A  e.  { B }  ->  A  =  B )
65orim2i 688 . 2  |-  ( ( A  e.  B  \/  A  e.  { B } )  ->  ( A  e.  B  \/  A  =  B )
)
74, 6sylbi 118 1  |-  ( A  e.  suc  B  -> 
( A  e.  B  \/  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 639    = wceq 1259    e. wcel 1409    u. cun 2943   {csn 3403   suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-suc 4136
This theorem is referenced by:  trsucss  4188  onsucelsucexmid  4283  ordsoexmid  4314  ordsuc  4315  ordpwsucexmid  4322  nnsucelsuc  6101  nntri3or  6103  nnmordi  6120  nnaordex  6131  phplem3  6348
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