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Theorem elsuci 4325
 Description: Membership in a successor. This one-way implication does not require that either or be sets. (Contributed by NM, 6-Jun-1994.)
Assertion
Ref Expression
elsuci

Proof of Theorem elsuci
StepHypRef Expression
1 df-suc 4293 . . . 4
21eleq2i 2206 . . 3
3 elun 3217 . . 3
42, 3bitri 183 . 2
5 elsni 3545 . . 3
65orim2i 750 . 2
74, 6sylbi 120 1
 Colors of variables: wff set class Syntax hints:   wi 4   wo 697   wceq 1331   wcel 1480   cun 3069  csn 3527   csuc 4287 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-suc 4293 This theorem is referenced by:  trsucss  4345  onsucelsucexmid  4445  ordsoexmid  4477  ordsuc  4478  ordpwsucexmid  4485  nnsucelsuc  6387  nntri3or  6389  nnmordi  6412  nnaordex  6423  phplem3  6748
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