Theorem elsuc2 3045

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 3043	. 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 146   \/ wo 222   = wceq 958   e. wcel 960  Vcvv 1814  suc csuc 2956

This theorem is referenced by:  alephordi 4885

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053  df-sn 2416  df-pr 2417  df-suc 2960

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