Theorem elsuc2 3043

Description: Membership in a successor.

Hypothesis

Ref	Expression
elsuc.1	|- A e. V

Assertion

Ref	Expression
elsuc2	|- (B e. suc A <-> (B e. A \/ B = A))

Proof of Theorem elsuc2

Step	Hyp	Ref	Expression
1		elsuc.1	. 2 |- A e. V
2		elsuc2g 3041	. 2 |- (A e. V -> (B e. suc A <-> (B e. A \/ B = A)))
3	1, 2	ax-mp 7	1 |- (B e. suc A <-> (B e. A \/ B = A))

Syntax hints:   <-> wb 144   \/ wo 220   = wceq 992   e. wcel 994  Vcvv 1857  suc csuc 2977

This theorem is referenced by:  alephordi 5024

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858  df-un 2102  df-sn 2470  df-pr 2471  df-suc 2981

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