Theorem sucid 4347

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

Step	Hyp	Ref	Expression
1		sucid.1	. 2 |- A e. _V
2		sucidg 4346	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 5	1 |- A e. suc A

Syntax hints:   e. wcel 1481   _Vcvv 2689   suc csuc 4295

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-suc 4301

This theorem is referenced by:  eqelsuc  4349  unon  4435  ordunisuc2r  4438  ordsoexmid  4485  limom  4535  0elnn  4540  tfrexlem  6239  tfri1dALT  6256  tfrcl  6269  frecabcl  6304  phplem4  6757  fiintim  6825  fidcenumlemr  6851  infnninf  7030  nnnninf  7031  prarloclemarch2  7251  prarloclemlt  7325  ennnfonelemex  11963  ennnfonelemrn  11968  bj-nn0suc0  13319  bj-nnelirr  13322  bj-inf2vnlem2  13340  bj-findis  13348  nninfsellemeq  13385