Theorem elsuc2g 3037

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 2436	. . . 4 |- (B e. C -> (A e. {B} <-> A = B))
2	1	orbi2d 614	. . 3 |- (B e. C -> ((A e. B \/ A e. {B}) <-> (A e. B \/ A = B)))
3		elun 2173	. . 3 |- (A e. (B u. {B}) <-> (A e. B \/ A e. {B}))
4	2, 3	syl5bb 532	. 2 |- (B e. C -> (A e. (B u. {B}) <-> (A e. B \/ A = B)))
5		df-suc 2954	. . 3 |- suc B = (B u. {B})
6	5	eleq2i 1538	. 2 |- (A e. suc B <-> A e. (B u. {B}))
7	4, 6	syl5bb 532	1 |- (B e. C -> (A e. suc B <-> (A e. B \/ A = B)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   = wceq 956   e. wcel 958   u. cun 2045  {csn 2409  suc csuc 2950

This theorem is referenced by:  elsuc2 3039  om2uzlt 6298

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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