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Theorem sucidg 4418
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 2177 . . 3 𝐴 = 𝐴
21olci 732 . 2 (𝐴𝐴𝐴 = 𝐴)
3 elsucg 4406 . 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 708   = wceq 1353  wcel 2148  suc csuc 4367
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-sn 3600  df-suc 4373
This theorem is referenced by:  sucid  4419  nsuceq0g  4420  trsuc  4424  sucssel  4426  ordsucg  4503  sucunielr  4511  suc11g  4558  nlimsucg  4567  peano2b  4616  omsinds  4623  nnpredlt  4625  frecsuclem  6409  phplem4dom  6864  phplem4on  6869  dif1en  6881  fin0  6887  fin0or  6888  fidcenumlemrks  6954  bj-peano4  14746
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