Theorem sucid 4235

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 4234	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 7	1 |- A e. suc A

Syntax hints:   e. wcel 1438   _Vcvv 2619   suc csuc 4183

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-suc 4189

This theorem is referenced by:  eqelsuc  4237  unon  4318  ordunisuc2r  4321  ordsoexmid  4368  limom  4418  0elnn  4422  tfrexlem  6081  tfri1dALT  6098  tfrcl  6111  frecabcl  6146  phplem4  6551  fidcenumlemr  6643  infnninf  6784  nnnninf  6785  prarloclemarch2  6957  prarloclemlt  7031  bj-nn0suc0  11502  bj-nnelirr  11505  bj-inf2vnlem2  11523  bj-findis  11531  nninfsellemeq  11563