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Theorem sucidg 4437
Description: Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). (Contributed by NM, 25-Mar-1995.) (Proof shortened by Scott Fenton, 20-Feb-2012.)
Assertion
Ref Expression
sucidg (𝐴𝑉𝐴 ∈ suc 𝐴)

Proof of Theorem sucidg
StepHypRef Expression
1 eqid 2189 . . 3 𝐴 = 𝐴
21olci 733 . 2 (𝐴𝐴𝐴 = 𝐴)
3 elsucg 4425 . 2 (𝐴𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴𝐴𝐴 = 𝐴)))
42, 3mpbiri 168 1 (𝐴𝑉𝐴 ∈ suc 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 709   = wceq 1364  wcel 2160  suc csuc 4386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3616  df-suc 4392
This theorem is referenced by:  sucid  4438  nsuceq0g  4439  trsuc  4443  sucssel  4445  ordsucg  4522  sucunielr  4530  suc11g  4577  nlimsucg  4586  peano2b  4635  omsinds  4642  nnpredlt  4644  frecsuclem  6435  phplem4dom  6894  phplem4on  6899  dif1en  6911  fin0  6917  fin0or  6918  fidcenumlemrks  6986  bj-peano4  15193
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