Theorem sucid 4482

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 4481	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 5	1 |- A e. suc A

Syntax hints:   e. wcel 2178   _Vcvv 2776   suc csuc 4430

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-suc 4436

This theorem is referenced by:  eqelsuc  4484  unon  4577  ordunisuc2r  4580  ordsoexmid  4628  limom  4680  0elnn  4685  tfrexlem  6443  tfri1dALT  6460  tfrcl  6473  frecabcl  6508  phplem4  6977  fiintim  7054  fidcenumlemr  7083  nninfwlpoimlemginf  7304  pw1ne3  7376  sucpw1ne3  7378  sucpw1nel3  7379  prarloclemarch2  7567  prarloclemlt  7641  ennnfonelemex  12900  ennnfonelemrn  12905  bj-nn0suc0  16085  bj-nnelirr  16088  bj-inf2vnlem2  16106  bj-findis  16114  nninfsellemeq  16153