Theorem sucid 4339

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 4338	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 5	1 |- A e. suc A

Syntax hints:   e. wcel 1480   _Vcvv 2686   suc 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:  eqelsuc  4341  unon  4427  ordunisuc2r  4430  ordsoexmid  4477  limom  4527  0elnn  4532  tfrexlem  6231  tfri1dALT  6248  tfrcl  6261  frecabcl  6296  phplem4  6749  fiintim  6817  fidcenumlemr  6843  infnninf  7022  nnnninf  7023  prarloclemarch2  7227  prarloclemlt  7301  ennnfonelemex  11927  ennnfonelemrn  11932  bj-nn0suc0  13148  bj-nnelirr  13151  bj-inf2vnlem2  13169  bj-findis  13177  nninfsellemeq  13210