Theorem sucid 4540

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 4539	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 5	1 |- A e. suc A

Syntax hints:   e. wcel 2205   _Vcvv 2815   suc csuc 4488

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217  df-sn 3697  df-suc 4494

This theorem is referenced by:  eqelsuc  4542  unon  4635  ordunisuc2r  4638  ordsoexmid  4686  limom  4738  0elnn  4743  tfrexlem  6567  tfri1dALT  6584  tfrcl  6597  frecabcl  6632  phplem4  7111  fiintim  7193  fidcenumlemr  7227  nninfwlpoimlemginf  7469  pw1ne3  7542  sucpw1ne3  7544  sucpw1nel3  7545  prarloclemarch2  7736  prarloclemlt  7810  ennnfonelemex  13182  ennnfonelemrn  13187  bj-nn0suc0  16737  bj-nnelirr  16740  bj-inf2vnlem2  16758  bj-findis  16766  3dom  16779  nninfsellemeq  16809