Theorem elsucg 3031

Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17.

Assertion

Ref	Expression
elsucg	|- (A e. C -> (A e. suc B <-> (A e. B \/ A = B)))

Proof of Theorem elsucg

Step	Hyp	Ref	Expression
1		elsncg 2426	. . 3 |- (A e. C -> (A e. {B} <-> A = B))
2	1	orbi2d 613	. 2 |- (A e. C -> ((A e. B \/ A e. {B}) <-> (A e. B \/ A = B)))
3		df-suc 2949	. . . 4 |- suc B = (B u. {B})
4	3	eleq2i 1535	. . 3 |- (A e. suc B <-> A e. (B u. {B}))
5		elun 2169	. . 3 |- (A e. (B u. {B}) <-> (A e. B \/ A e. {B}))
6	4, 5	bitr 173	. 2 |- (A e. suc B <-> (A e. B \/ A e. {B}))
7	2, 6	syl5bb 531	1 |- (A e. C -> (A e. suc B <-> (A e. B \/ A = B)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   = wceq 954   e. wcel 956   u. cun 2041  {csn 2405  suc csuc 2945

This theorem is referenced by:  elsuc 3033  elelsuc 3036  ordsssuc 3052  ordsucelsuc 3068  suc11reg 4585  nlt1pi 5013

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409  df-suc 2949

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