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Theorem sucid 4413
Description: A set belongs to its successor. (Contributed by NM, 22-Jun-1994.) (Proof shortened by Alan Sare, 18-Feb-2012.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
Hypothesis
Ref Expression
sucid.1  |-  A  e. 
_V
Assertion
Ref Expression
sucid  |-  A  e. 
suc  A

Proof of Theorem sucid
StepHypRef Expression
1 sucid.1 . 2  |-  A  e. 
_V
2 sucidg 4412 . 2  |-  ( A  e.  _V  ->  A  e.  suc  A )
31, 2ax-mp 5 1  |-  A  e. 
suc  A
Colors of variables: wff set class
Syntax hints:    e. wcel 2148   _Vcvv 2737   suc csuc 4361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-sn 3597  df-suc 4367
This theorem is referenced by:  eqelsuc  4415  unon  4506  ordunisuc2r  4509  ordsoexmid  4557  limom  4609  0elnn  4614  tfrexlem  6328  tfri1dALT  6345  tfrcl  6358  frecabcl  6393  phplem4  6848  fiintim  6921  fidcenumlemr  6947  nninfwlpoimlemginf  7167  pw1ne3  7222  sucpw1ne3  7224  sucpw1nel3  7225  prarloclemarch2  7396  prarloclemlt  7470  ennnfonelemex  12385  ennnfonelemrn  12390  bj-nn0suc0  14324  bj-nnelirr  14327  bj-inf2vnlem2  14345  bj-findis  14353  nninfsellemeq  14386
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