Theorem sucid 4557

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 4556	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 5	1 |- A e. suc A

Syntax hints:   e. wcel 2209   _Vcvv 2821   suc csuc 4505

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3711  df-suc 4511

This theorem is referenced by:  eqelsuc  4559  unon  4653  ordunisuc2r  4656  ordsoexmid  4704  limom  4756  0elnn  4761  tfrexlem  6595  tfri1dALT  6612  tfrcl  6625  frecabcl  6660  phplem4  7146  fiintim  7228  fidcenumlemr  7262  nninfwlpoimlemginf  7506  pw1ne3  7579  sucpw1ne3  7581  sucpw1nel3  7582  prarloclemarch2  7776  prarloclemlt  7850  ennnfonelemex  13283  ennnfonelemrn  13288  bj-nn0suc0  16890  bj-nnelirr  16893  bj-inf2vnlem2  16911  bj-findis  16919  3dom  16932  nninfsellemeq  16962