MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sucidg Structured version   Visualization version   GIF version

Theorem sucidg 5772
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 5761 . 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 5694
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 3192  df-un 3565  df-sn 4156  df-suc 5698
This theorem is referenced by:  sucid  5773  nsuceq0  5774  trsuc  5779  sucssel  5788  ordsuc  6976  onpsssuc  6981  nlimsucg  7004  tfrlem11  7444  tfrlem13  7446  tz7.44-2  7463  omeulem1  7622  oeordi  7627  oeeulem  7641  php4  8107  wofib  8410  suc11reg  8476  cantnfle  8528  cantnflt2  8530  cantnfp1lem3  8537  cantnflem1  8546  dfac12lem1  8925  dfac12lem2  8926  ttukeylem3  9293  ttukeylem7  9297  r1wunlim  9519  noreslege  31624  noprefixmo  31626  ontgval  32125  sucneqond  32884  finxpreclem4  32902  finxpsuclem  32905
  Copyright terms: Public domain W3C validator