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Theorem sucidg 4338
 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 2139 . . 3 𝐴 = 𝐴
21olci 721 . 2 (𝐴𝐴𝐴 = 𝐴)
3 elsucg 4326 . 2 (𝐴𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴𝐴𝐴 = 𝐴)))
42, 3mpbiri 167 1 (𝐴𝑉𝐴 ∈ suc 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∨ wo 697   = wceq 1331   ∈ wcel 1480  suc csuc 4287 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-sn 3533  df-suc 4293 This theorem is referenced by:  sucid  4339  nsuceq0g  4340  trsuc  4344  sucssel  4346  ordsucg  4418  sucunielr  4426  suc11g  4472  nlimsucg  4481  peano2b  4528  omsinds  4535  frecsuclem  6303  phplem4dom  6756  phplem4on  6761  dif1en  6773  fin0  6779  fin0or  6780  fidcenumlemrks  6841  bj-peano4  13212
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