Theorem elsuc2g 3043

Description: Variant of membership in a successor, requiring that B rather than A be a set.

Assertion

Ref	Expression
elsuc2g	|- (B e. C -> (A e. suc B <-> (A e. B \/ A = B)))

Proof of Theorem elsuc2g

Step	Hyp	Ref	Expression
1		elsnc2g 2440	. . . 4 |- (B e. C -> (A e. {B} <-> A = B))
2	1	orbi2d 616	. . 3 |- (B e. C -> ((A e. B \/ A e. {B}) <-> (A e. B \/ A = B)))
3		elun 2176	. . 3 |- (A e. (B u. {B}) <-> (A e. B \/ A e. {B}))
4	2, 3	syl5bb 534	. 2 |- (B e. C -> (A e. (B u. {B}) <-> (A e. B \/ A = B)))
5		df-suc 2960	. . 3 |- suc B = (B u. {B})
6	5	eleq2i 1541	. 2 |- (A e. suc B <-> A e. (B u. {B}))
7	4, 6	syl5bb 534	1 |- (B e. C -> (A e. suc B <-> (A e. B \/ A = B)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   = wceq 958   e. wcel 960   u. cun 2048  {csn 2413  suc csuc 2956

This theorem is referenced by:  elsuc2 3045  om2uzlt 6299

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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