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

Theorem sucex 7826
Description: The successor of a set is a set. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
sucex.1 𝐴 ∈ V
Assertion
Ref Expression
sucex suc 𝐴 ∈ V

Proof of Theorem sucex
StepHypRef Expression
1 sucex.1 . 2 𝐴 ∈ V
2 sucexg 7825 . 2 (𝐴 ∈ V → suc 𝐴 ∈ V)
31, 2ax-mp 5 1 suc 𝐴 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  Vcvv 3478  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-suc 6392
This theorem is referenced by:  orduninsuc  7864  tfindsg  7882  tfinds2  7885  finds  7919  findsg  7920  finds2  7921  seqomlem1  8489  oasuc  8561  onasuc  8565  naddcllem  8713  infensuc  9194  phplem4OLD  9255  phpOLD  9257  inf0  9659  inf3lem1  9666  dfom3  9685  cantnflt  9710  cantnflem1  9727  cnfcom  9738  brttrcl2  9752  ssttrcl  9753  ttrcltr  9754  ttrclss  9758  ttrclselem2  9764  infxpenlem  10051  pwsdompw  10241  cfslb2n  10306  cfsmolem  10308  fin1a2lem12  10449  axdc4lem  10493  alephreg  10620  bnj986  34948  bnj1018g  34956  bnj1018  34957  satf  35338  dfon2lem7  35771  rdgssun  37361  dford3lem2  43016
  Copyright terms: Public domain W3C validator