Theorem sucid 4465

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 4464	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 5	1 |- A e. suc A

Syntax hints:   e. wcel 2176   _Vcvv 2772   suc csuc 4413

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-suc 4419

This theorem is referenced by:  eqelsuc  4467  unon  4560  ordunisuc2r  4563  ordsoexmid  4611  limom  4663  0elnn  4668  tfrexlem  6422  tfri1dALT  6439  tfrcl  6452  frecabcl  6487  phplem4  6954  fiintim  7030  fidcenumlemr  7059  nninfwlpoimlemginf  7280  pw1ne3  7344  sucpw1ne3  7346  sucpw1nel3  7347  prarloclemarch2  7534  prarloclemlt  7608  ennnfonelemex  12818  ennnfonelemrn  12823  bj-nn0suc0  15923  bj-nnelirr  15926  bj-inf2vnlem2  15944  bj-findis  15952  nninfsellemeq  15988