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Theorem sucidg 6465
Description: Part of Proposition 7.23 of [TakeutiZaring] p. 41 (generalized). Lemma 1.7 of [Schloeder] p. 1. (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 2737 . . 3 𝐴 = 𝐴
21olci 867 . 2 (𝐴𝐴𝐴 = 𝐴)
3 elsucg 6452 . 2 (𝐴𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴𝐴𝐴 = 𝐴)))
42, 3mpbiri 258 1 (𝐴𝑉𝐴 ∈ suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 848   = wceq 1540  wcel 2108  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-sn 4627  df-suc 6390
This theorem is referenced by:  sucid  6466  nsuceq0  6467  trsuc  6471  sucssel  6479  ordsuc  7833  ordsucOLD  7834  onpsssuc  7839  nlimsucg  7863  tfrlem11  8428  tfrlem13  8430  tz7.44-2  8447  omeulem1  8620  oeordi  8625  oeeulem  8639  dif1enlem  9196  dif1enlemOLD  9197  rexdif1en  9198  rexdif1enOLD  9199  dif1en  9200  dif1enOLD  9202  php4  9250  wofib  9585  suc11reg  9659  cantnfle  9711  cantnflt2  9713  cantnfp1lem3  9720  cantnflem1  9729  dfac12lem1  10184  dfac12lem2  10185  ttukeylem3  10551  ttukeylem7  10555  r1wunlim  10777  noresle  27742  nosupprefixmo  27745  noinfprefixmo  27746  fmla  35386  ex-sategoelelomsuc  35431  ontgval  36432  sucneqond  37366  finxpreclem4  37395  finxpsuclem  37398  onexgt  43252  onepsuc  43264  ordnexbtwnsuc  43280  nlimsuc  43454  sucomisnotcard  43557
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