Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sucel Structured version   Unicode version

Theorem sucel 5453
 Description: Membership of a successor in another class. (Contributed by NM, 29-Jun-2004.)
Assertion
Ref Expression
sucel
Distinct variable groups:   ,,   ,
Allowed substitution hint:   ()

Proof of Theorem sucel
StepHypRef Expression
1 risset 2887 . 2
2 dfcleq 2417 . . . 4
3 vex 3020 . . . . . . 7
43elsuc 5449 . . . . . 6
54bibi2i 314 . . . . 5
65albii 1685 . . . 4
72, 6bitri 252 . . 3
87rexbii 2861 . 2
91, 8bitri 252 1
 Colors of variables: wff setvar class Syntax hints:   wb 187   wo 369  wal 1435   wceq 1437   wcel 1872  wrex 2710   csuc 5382 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-rex 2715  df-v 3019  df-un 3379  df-sn 3937  df-suc 5386 This theorem is referenced by:  axinf2  8093  zfinf2  8095
 Copyright terms: Public domain W3C validator