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Theorem sucidg 4412
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 4400 . 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 4361
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 2739  df-un 3133  df-sn 3597  df-suc 4367
This theorem is referenced by:  sucid  4413  nsuceq0g  4414  trsuc  4418  sucssel  4420  ordsucg  4497  sucunielr  4505  suc11g  4552  nlimsucg  4561  peano2b  4610  omsinds  4617  nnpredlt  4619  frecsuclem  6400  phplem4dom  6855  phplem4on  6860  dif1en  6872  fin0  6878  fin0or  6879  fidcenumlemrks  6945  bj-peano4  14329
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