Theorem sucid 3744

Description: A set belongs to its successor. (The proof was shortened by Scott Fenton, 18-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 3743	. 2 |- (A e. _V -> A e. suc A)
3	1, 2	ax-mp 7	1 |- A e. suc A

Syntax hints:   e. wcel 1300  _Vcvv 2292  suc csuc 3659

This theorem is referenced by:  sucidgOLD 3748  eqelsuc 3750  unon 3910  onuninsuci 3921  tfinds 3942  tfindsOLD 3943  peano5 3975  tz7.44-2 5137  oawordeulem 5236  oalimcl 5242  omlimcl 5257  oneo 5260  oeworde 5268  ac6sfilem2 5507  ac6sfilem3 5508  ac6sfi 5509  phplem4 5605  php 5607  unifi 5648  fiint 5650  fodomfi 5656  inf0 5712  oancom 5740  r1val1 5769  rankwflem 5776  rankr1 5785  rankxplim3 5825  infenomsub 5889  cardlim 6003  cardaleph 6033  fbssint 10279  bnj216 12507  bnj98 13221  axfelem15 14045  finsschain 15373  infenomsubOLD 15398  fcluscomplem 15620  smoge 16454

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-suc 3663

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