Theorem sucid 4377

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 4376	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 5	1 |- A e. suc A

Syntax hints:   e. wcel 2128   _Vcvv 2712   suc csuc 4325

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-sn 3566  df-suc 4331

This theorem is referenced by:  eqelsuc  4379  unon  4469  ordunisuc2r  4472  ordsoexmid  4520  limom  4572  0elnn  4577  tfrexlem  6278  tfri1dALT  6295  tfrcl  6308  frecabcl  6343  phplem4  6797  fiintim  6870  fidcenumlemr  6896  pw1ne3  7159  sucpw1ne3  7161  sucpw1nel3  7162  prarloclemarch2  7333  prarloclemlt  7407  ennnfonelemex  12126  ennnfonelemrn  12131  bj-nn0suc0  13496  bj-nnelirr  13499  bj-inf2vnlem2  13517  bj-findis  13525  nninfsellemeq  13557