Theorem sucid 6425

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 ∈ V

Assertion

Ref	Expression
sucid	⊢ 𝐴 ∈ suc 𝐴

Proof of Theorem sucid

Step	Hyp	Ref	Expression
1		sucid.1	. 2 ⊢ 𝐴 ∈ V
2		sucidg 6424	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2141  Vcvv 3453  suc csuc 6343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3907  df-sn 4580  df-suc 6347

This theorem is referenced by:  eqelsuc  6427  unon  7806  onuninsuci  7815  tfinds  7835  peano5  7869  tfrlem16  8358  oawordeulem  8517  oalimcl  8523  omlimcl  8541  oneo  8544  omeulem1  8545  oeworde  8557  nnawordex  8601  nnneo  8619  naddcllem  8640  phplem2  9167  php  9169  fiint  9265  inf0  9570  oancom  9600  cantnfval2  9618  cantnflt  9621  cantnflem1  9638  cnfcom  9649  cnfcom2  9651  cnfcom3lem  9652  cnfcom3  9653  ssttrcl  9664  ttrcltr  9665  ttrclss  9669  rnttrcl  9671  ttrclselem2  9675  r1val1  9738  rankxplim3  9833  cardlim  9924  fseqenlem1  9974  cardaleph  10039  pwsdompw  10153  cfsmolem  10221  axdc3lem4  10404  ttukeylem5  10464  ttukeylem6  10465  ttukeylem7  10466  canthp1lem2  10605  pwxpndom2  10617  winainflem  10645  winalim2  10648  nqereu  10881  nogt01o  27748  bdayiun  27996  n0bday  28433  bnj216  34989  bnj98  35123  fineqvnttrclse  35381  satom  35667  fmla  35692  ex-sategoelel12  35738  dfrdg2  36104  nmulprop  36501  preel  38960  dford3lem2  43565  pw2f1ocnv  43575  aomclem1  43592  nnoeomeqom  43850  naddgeoa  43932  naddwordnexlem4  43939