Theorem sucid 6385

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 6384	. 2 ⊢ (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
3	1, 2	ax-mp 5	1 ⊢ 𝐴 ∈ suc 𝐴

Syntax hints:   ∈ wcel 2111  Vcvv 3436  suc csuc 6303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-sn 4572  df-suc 6307

This theorem is referenced by:  eqelsuc  6387  unon  7756  onuninsuci  7765  tfinds  7785  peano5  7818  tfrlem16  8307  oawordeulem  8464  oalimcl  8470  omlimcl  8488  oneo  8491  omeulem1  8492  oeworde  8503  nnawordex  8547  nnneo  8565  naddcllem  8586  phplem2  9109  php  9111  fiint  9206  inf0  9506  oancom  9536  cantnfval2  9554  cantnflt  9557  cantnflem1  9574  cnfcom  9585  cnfcom2  9587  cnfcom3lem  9588  cnfcom3  9589  ssttrcl  9600  ttrcltr  9601  ttrclss  9605  rnttrcl  9607  ttrclselem2  9611  r1val1  9674  rankxplim3  9769  cardlim  9860  fseqenlem1  9910  cardaleph  9975  pwsdompw  10089  cfsmolem  10156  axdc3lem4  10339  ttukeylem5  10399  ttukeylem6  10400  ttukeylem7  10401  canthp1lem2  10539  pwxpndom2  10551  winainflem  10579  winalim2  10582  nqereu  10815  nogt01o  27630  bdayiun  27855  n0sbday  28275  bnj216  34736  bnj98  34871  fineqvnttrclse  35136  satom  35392  fmla  35417  ex-sategoelel12  35463  dfrdg2  35829  dford3lem2  43060  pw2f1ocnv  43070  aomclem1  43087  nnoeomeqom  43345  naddgeoa  43427  naddwordnexlem4  43434