Theorem sucid 4512

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 4511	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 5	1 |- A e. suc A

Syntax hints:   e. wcel 2200   _Vcvv 2800   suc csuc 4460

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202  df-sn 3673  df-suc 4466

This theorem is referenced by:  eqelsuc  4514  unon  4607  ordunisuc2r  4610  ordsoexmid  4658  limom  4710  0elnn  4715  tfrexlem  6495  tfri1dALT  6512  tfrcl  6525  frecabcl  6560  phplem4  7036  fiintim  7116  fidcenumlemr  7145  nninfwlpoimlemginf  7366  pw1ne3  7438  sucpw1ne3  7440  sucpw1nel3  7441  prarloclemarch2  7629  prarloclemlt  7703  ennnfonelemex  13025  ennnfonelemrn  13030  bj-nn0suc0  16481  bj-nnelirr  16484  bj-inf2vnlem2  16502  bj-findis  16510  3dom  16523  nninfsellemeq  16552