Theorem sucid 4464

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 4463	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 5	1 |- A e. suc A

Syntax hints:   e. wcel 2176   _Vcvv 2772   suc csuc 4412

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-suc 4418

This theorem is referenced by:  eqelsuc  4466  unon  4559  ordunisuc2r  4562  ordsoexmid  4610  limom  4662  0elnn  4667  tfrexlem  6420  tfri1dALT  6437  tfrcl  6450  frecabcl  6485  phplem4  6952  fiintim  7028  fidcenumlemr  7057  nninfwlpoimlemginf  7278  pw1ne3  7342  sucpw1ne3  7344  sucpw1nel3  7345  prarloclemarch2  7532  prarloclemlt  7606  ennnfonelemex  12785  ennnfonelemrn  12790  bj-nn0suc0  15886  bj-nnelirr  15889  bj-inf2vnlem2  15907  bj-findis  15915  nninfsellemeq  15951