Theorem elsucg 3040

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 2491	. . 3 |- (A e. C -> (A e. {B} <-> A = B))
2	1	orbi2d 617	. 2 |- (A e. C -> ((A e. B \/ A e. {B}) <-> (A e. B \/ A = B)))
3		df-suc 2981	. . . 4 |- suc B = (B u. {B})
4	3	eleq2i 1581	. . 3 |- (A e. suc B <-> A e. (B u. {B}))
5		elun 2225	. . 3 |- (A e. (B u. {B}) <-> (A e. B \/ A e. {B}))
6	4, 5	bitri 171	. 2 |- (A e. suc B <-> (A e. B \/ A e. {B}))
7	2, 6	syl5bb 535	1 |- (A e. C -> (A e. suc B <-> (A e. B \/ A = B)))

Syntax hints:   -> wi 3   <-> wb 144   \/ wo 220   = wceq 992   e. wcel 994   u. cun 2097  {csn 2467  suc csuc 2977

This theorem is referenced by:  elsuc 3042  elelsuc 3045  ordsssuc 3058  ordsucelsuc 3178  suc11reg 4750  nlt1pi 5187

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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