Theorem sucid 3051

Description: A set belongs to its successor.

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	. . 3 |- A e. V
2	1	snid 2435	. 2 |- A e. {A}
3		df-suc 2954	. . . . . 6 |- suc A = (A u. {A})
4	3	eleq2i 1538	. . . . 5 |- (A e. suc A <-> A e. (A u. {A}))
5		elun 2173	. . . . 5 |- (A e. (A u. {A}) <-> (A e. A \/ A e. {A}))
6	4, 5	bitr 173	. . . 4 |- (A e. suc A <-> (A e. A \/ A e. {A}))
7	6	biimpr 152	. . 3 |- ((A e. A \/ A e. {A}) -> A e. suc A)
8	7	olcs 275	. 2 |- (A e. {A} -> A e. suc A)
9	2, 8	ax-mp 7	1 |- A e. suc A

Syntax hints:   \/ wo 222   e. wcel 958  Vcvv 1811   u. cun 2045  {csn 2409  suc csuc 2950

This theorem is referenced by:  sucidg 3052  eqelsuc 3054  unon 3088  onuninsuc 3108  peano5 3153  tfinds 3161  tz7.44-2 3929  oawordeulem 4188  oalimcl 4194  omlimcl 4209  oneo 4212  oeworde 4220  phplem4 4511  php 4513  unifiOLD 4557  fiint 4559  fiintOLD 4560  fodomfiOLD 4566  inf0 4606  oancom 4633  r1val1 4658  rankwflem 4665  rankr1 4674  rankxplim3 4714  cardlim 4851  cardaleph 4885

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-sn 2412  df-pr 2413  df-suc 2954

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