Theorem sucid 4453

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 4452	. 2 |- ( A e. _V -> A e. suc A )
3	1, 2	ax-mp 5	1 |- A e. suc A

Syntax hints:   e. wcel 2167   _Vcvv 2763   suc csuc 4401

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3629  df-suc 4407

This theorem is referenced by:  eqelsuc  4455  unon  4548  ordunisuc2r  4551  ordsoexmid  4599  limom  4651  0elnn  4656  tfrexlem  6401  tfri1dALT  6418  tfrcl  6431  frecabcl  6466  phplem4  6925  fiintim  7001  fidcenumlemr  7030  nninfwlpoimlemginf  7251  pw1ne3  7313  sucpw1ne3  7315  sucpw1nel3  7316  prarloclemarch2  7503  prarloclemlt  7577  ennnfonelemex  12656  ennnfonelemrn  12661  bj-nn0suc0  15680  bj-nnelirr  15683  bj-inf2vnlem2  15701  bj-findis  15709  nninfsellemeq  15745