Theorem sucid 4435

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 4434	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 5	1 |- A e. suc A

Syntax hints:   e. wcel 2160   _Vcvv 2752   suc csuc 4383

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-suc 4389

This theorem is referenced by:  eqelsuc  4437  unon  4528  ordunisuc2r  4531  ordsoexmid  4579  limom  4631  0elnn  4636  tfrexlem  6360  tfri1dALT  6377  tfrcl  6390  frecabcl  6425  phplem4  6884  fiintim  6958  fidcenumlemr  6985  nninfwlpoimlemginf  7205  pw1ne3  7260  sucpw1ne3  7262  sucpw1nel3  7263  prarloclemarch2  7449  prarloclemlt  7523  ennnfonelemex  12468  ennnfonelemrn  12473  bj-nn0suc0  15180  bj-nnelirr  15183  bj-inf2vnlem2  15201  bj-findis  15209  nninfsellemeq  15242