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Theorem elsucg 4090
 Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
Assertion
Ref Expression
elsucg (A 𝑉 → (A suc B ↔ (A B A = B)))

Proof of Theorem elsucg
StepHypRef Expression
1 df-suc 4057 . . . 4 suc B = (B ∪ {B})
21eleq2i 2086 . . 3 (A suc BA (B ∪ {B}))
3 elun 3061 . . 3 (A (B ∪ {B}) ↔ (A B A {B}))
42, 3bitri 173 . 2 (A suc B ↔ (A B A {B}))
5 elsncg 3372 . . 3 (A 𝑉 → (A {B} ↔ A = B))
65orbi2d 691 . 2 (A 𝑉 → ((A B A {B}) ↔ (A B A = B)))
74, 6syl5bb 181 1 (A 𝑉 → (A suc B ↔ (A B A = B)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∨ wo 616   = wceq 1228   ∈ wcel 1374   ∪ cun 2892  {csn 3350  suc csuc 4051 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004 This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-sn 3356  df-suc 4057 This theorem is referenced by:  elsuc  4092  elelsuc  4095  sucidg  4102  onsucelsucr  4183  onsucsssucexmid  4196  suc11g  4219  nlt1pig  6201  bj-peano4  7177
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