Theorem elsuci 3039

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 2981	. . . 4 |- suc B = (B u. {B})
2	1	eleq2i 1581	. . 3 |- (A e. suc B <-> A e. (B u. {B}))
3		elun 2225	. . 3 |- (A e. (B u. {B}) <-> (A e. B \/ A e. {B}))
4	2, 3	bitri 171	. 2 |- (A e. suc B <-> (A e. B \/ A e. {B}))
5		elsni 2493	. . 3 |- (A e. {B} -> A = B)
6	5	orim2i 336	. 2 |- ((A e. B \/ A e. {B}) -> (A e. B \/ A = B))
7	4, 6	sylbi 197	1 |- (A e. suc B -> (A e. B \/ A = B))

Syntax hints:   -> wi 3   \/ wo 220   = wceq 992   e. wcel 994   u. cun 2097  {csn 2467  suc csuc 2977

This theorem is referenced by:  trsucss 3057  ordnbtwn 3062  suc11 3073  tfrlem11 4222  omordi 4333  phplem3 4657  pssnn 4681  cfsuc 5065  indpi 5188  elsucii 12212

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-v 1858  df-un 2102  df-sn 2470  df-pr 2471  df-suc 2981

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