Theorem elsuci 3035

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 2954	. . . 4 |- suc B = (B u. {B})
2	1	eleq2i 1538	. . 3 |- (A e. suc B <-> A e. (B u. {B}))
3		elun 2173	. . 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 2432	. . 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 956   e. wcel 958   u. cun 2045  {csn 2409  suc csuc 2950

This theorem is referenced by:  trsucss 3056  ordnbtwn 3063  suc11 3093  tfrlem11 3921  omordi 4197  phplem3 4510  pssnn 4534  cfsuc 4915  indpi 5034

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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