Theorem sucid 4538

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 4537	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 5	1 |- A e. suc A

Syntax hints:   e. wcel 2203   _Vcvv 2813   suc csuc 4486

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-sn 3695  df-suc 4492

This theorem is referenced by:  eqelsuc  4540  unon  4633  ordunisuc2r  4636  ordsoexmid  4684  limom  4736  0elnn  4741  tfrexlem  6565  tfri1dALT  6582  tfrcl  6595  frecabcl  6630  phplem4  7109  fiintim  7191  fidcenumlemr  7225  nninfwlpoimlemginf  7467  pw1ne3  7540  sucpw1ne3  7542  sucpw1nel3  7543  prarloclemarch2  7734  prarloclemlt  7808  ennnfonelemex  13165  ennnfonelemrn  13170  bj-nn0suc0  16720  bj-nnelirr  16723  bj-inf2vnlem2  16741  bj-findis  16749  3dom  16762  nninfsellemeq  16792