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Theorem sucidALT 37126
Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 37125 using translatewithout_overwriting.cmd and minimizing. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
sucidALT.1  |-  A  e. 
_V
Assertion
Ref Expression
sucidALT  |-  A  e. 
suc  A

Proof of Theorem sucidALT
StepHypRef Expression
1 sucidALT.1 . . . 4  |-  A  e. 
_V
21snid 4024 . . 3  |-  A  e. 
{ A }
3 elun1 3633 . . 3  |-  ( A  e.  { A }  ->  A  e.  ( { A }  u.  A
) )
42, 3ax-mp 5 . 2  |-  A  e.  ( { A }  u.  A )
5 df-suc 5444 . . 3  |-  suc  A  =  ( A  u.  { A } )
65equncomi 3612 . 2  |-  suc  A  =  ( { A }  u.  A )
74, 6eleqtrri 2509 1  |-  A  e. 
suc  A
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1868   _Vcvv 3081    u. cun 3434   {csn 3996   suc csuc 5440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-v 3083  df-un 3441  df-in 3443  df-ss 3450  df-sn 3997  df-suc 5444
This theorem is referenced by: (None)
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