Theorem elsuc2g 3733

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 3070	. . . 4 |- (B e. C -> (A e. {B} <-> A = B))
2	1	orbi2d 676	. . 3 |- (B e. C -> ((A e. B \/ A e. {B}) <-> (A e. B \/ A = B)))
3		elun 2741	. . 3 |- (A e. (B u. {B}) <-> (A e. B \/ A e. {B}))
4	2, 3	syl5bb 591	. 2 |- (B e. C -> (A e. (B u. {B}) <-> (A e. B \/ A = B)))
5		df-suc 3663	. . 3 |- suc B = (B u. {B})
6	5	eleq2i 1961	. 2 |- (A e. suc B <-> A e. (B u. {B}))
7	4, 6	syl5bb 591	1 |- (B e. C -> (A e. suc B <-> (A e. B \/ A = B)))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   = wceq 1298   e. wcel 1300   u. cun 2591  {csn 3044  suc csuc 3659

This theorem is referenced by:  elsuc2 3735  om2uzlti 7709

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-suc 3663

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