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Theorem sucidg 4180
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 2056 . . 3 𝐴 = 𝐴
21olci 661 . 2 (𝐴𝐴𝐴 = 𝐴)
3 elsucg 4168 . 2 (𝐴𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴𝐴𝐴 = 𝐴)))
42, 3mpbiri 161 1 (𝐴𝑉𝐴 ∈ suc 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 639   = wceq 1259  wcel 1409  suc csuc 4129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-sn 3408  df-suc 4135
This theorem is referenced by:  sucid  4181  nsuceq0g  4182  trsuc  4186  sucssel  4188  ordsucg  4255  sucunielr  4263  suc11g  4308  nlimsucg  4317  onpsssuc  4322  peano2b  4364  frecsuclem2  6019  phplem4dom  6354  phplem4on  6359  dif1en  6367  fin0  6372  fin0or  6373  bj-peano4  10439
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