Theorem sucidOLD 3745

Description: A set belongs to its successor. (The proof was shortened by Alan Sare, 18-Feb-2012.) This proof was automatically generated from sucidVD 16696 using translate_without_overwriting.cmd and minimizing.

Hypothesis

Ref	Expression
sucid.1	|- A e. _V

Assertion

Ref	Expression
sucidOLD	|- A e. suc A

Proof of Theorem sucidOLD

Step	Hyp	Ref	Expression
1		sucid.1	. . . 4 |- A e. _V
2	1	snid 3069	. . 3 |- A e. {A}
3		elun2 2772	. . 3 |- (A e. {A} -> A e. (A u. {A}))
4	2, 3	ax-mp 7	. 2 |- A e. (A u. {A})
5		df-suc 3663	. 2 |- suc A = (A u. {A})
6	4, 5	eleqtrri 1970	1 |- A e. suc A

Syntax hints:   e. wcel 1300  _Vcvv 2292   u. cun 2591  {csn 3044  suc csuc 3659

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-in 2603  df-ss 2605  df-sn 3049  df-suc 3663

Copyright terms: Public domain