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Theorem sucidg 5762
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 2621 . . 3 𝐴 = 𝐴
21olci 406 . 2 (𝐴𝐴𝐴 = 𝐴)
3 elsucg 5751 . 2 (𝐴𝑉 → (𝐴 ∈ suc 𝐴 ↔ (𝐴𝐴𝐴 = 𝐴)))
42, 3mpbiri 248 1 (𝐴𝑉𝐴 ∈ suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 383   = wceq 1480  wcel 1987  suc csuc 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3560  df-sn 4149  df-suc 5688
This theorem is referenced by:  sucid  5763  nsuceq0  5764  trsuc  5769  sucssel  5778  ordsuc  6961  onpsssuc  6966  nlimsucg  6989  tfrlem11  7429  tfrlem13  7431  tz7.44-2  7448  omeulem1  7607  oeordi  7612  oeeulem  7626  php4  8091  wofib  8394  suc11reg  8460  cantnfle  8512  cantnflt2  8514  cantnfp1lem3  8521  cantnflem1  8530  dfac12lem1  8909  dfac12lem2  8910  ttukeylem3  9277  ttukeylem7  9281  r1wunlim  9503  noreslege  31571  noprefixmo  31573  ontgval  32072  sucneqond  32845  finxpreclem4  32863  finxpsuclem  32866
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