Theorem elsuc 3038

Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17.

Hypothesis

Ref	Expression
elsuc.1	|- A e. V

Assertion

Ref	Expression
elsuc	|- (A e. suc B <-> (A e. B \/ A = B))

Proof of Theorem elsuc

Step	Hyp	Ref	Expression
1		elsuc.1	. 2 |- A e. V
2		elsucg 3036	. 2 |- (A e. V -> (A e. suc B <-> (A e. B \/ A = B)))
3	1, 2	ax-mp 7	1 |- (A e. suc B <-> (A e. B \/ A = B))

Syntax hints:   <-> wb 146   \/ wo 222   = wceq 956   e. wcel 958  Vcvv 1811  suc csuc 2950

This theorem is referenced by:  hbsuc 3040  sucel 3042  limsssuc 3121  oarec 4196  omsmolem 4256

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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