Theorem elsuci 3731

Description: Membership in a successor. This one-way implication does not require that either A or B be sets.

Assertion

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

Proof of Theorem elsuci

Step	Hyp	Ref	Expression
1		df-suc 3663	. . . 4 |- suc B = (B u. {B})
2	1	eleq2i 1961	. . 3 |- (A e. suc B <-> A e. (B u. {B}))
3		elun 2741	. . 3 |- (A e. (B u. {B}) <-> (A e. B \/ A e. {B}))
4	2, 3	bitri 190	. 2 |- (A e. suc B <-> (A e. B \/ A e. {B}))
5		elsni 3066	. . 3 |- (A e. {B} -> A = B)
6	5	orim2i 365	. 2 |- ((A e. B \/ A e. {B}) -> (A e. B \/ A = B))
7	4, 6	sylbi 216	1 |- (A e. suc B -> (A e. B \/ A = B))

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

This theorem is referenced by:  trsucss 3755  ordnbtwn 3761  suc11 3773  tfrlem11 5129  omordi 5245  phplem3 5604  pssnn 5628  cfsuc 6063  indpi 6186  bnj559 12539  bnj964 13347  suctrALT2VD 16660  suctrALT2 16661

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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