Theorem elsuci 3025

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 2944	. . . 4 |- suc B = (B u. {B})
2	1	eleq2i 1530	. . 3 |- (A e. suc B <-> A e. (B u. {B}))
3		elun 2163	. . 3 |- (A e. (B u. {B}) <-> (A e. B \/ A e. {B}))
4	2, 3	bitr 173	. 2 |- (A e. suc B <-> (A e. B \/ A e. {B}))
5		elsni 2422	. . 3 |- (A e. {B} -> A = B)
6	5	orim2i 338	. 2 |- ((A e. B \/ A e. {B}) -> (A e. B \/ A = B))
7	4, 6	sylbi 199	1 |- (A e. suc B -> (A e. B \/ A = B))

Syntax hints:   -> wi 3   \/ wo 222   = wceq 953   e. wcel 955   u. cun 2035  {csn 2399  suc csuc 2940

This theorem is referenced by:  trsucss 3046  ordnbtwn 3053  suc11 3083  tfrlem11 3906  omordi 4181  phplem3 4490  pssnn 4513  cfsuc 4887  indpi 5006

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-un 2040  df-sn 2402  df-pr 2403  df-suc 2944

Copyright terms: Public domain