Theorem sucidALT 39107

Description: A set belongs to its successor. This proof was automatically derived from sucidALTVD 39106 using translate_without_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

Step	Hyp	Ref	Expression
1		sucidALT.1	. . . 4 |- A e. _V
2	1	snid 4208	. . 3 |- A e. { A }
3		elun1 3780	. . 3 |- ( A e. { A } -> A e. ( { A } u. A ) )
4	2, 3	ax-mp 5	. 2 |- A e. ( { A } u. A )
5		df-suc 5729	. . 3 |- suc A = ( A u. { A } )
6	5	equncomi 3759	. 2 |- suc A = ( { A } u. A )
7	4, 6	eleqtrri 2700	1 |- A e. suc A

Syntax hints:   e. wcel 1990   _Vcvv 3200   u. cun 3572   {csn 4177   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-sn 4178  df-suc 5729

This theorem is referenced by: (None)