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

Theorem sucex 6973
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 6972 . 2 (𝐴 ∈ V → suc 𝐴 ∈ V)
31, 2ax-mp 5 1 suc 𝐴 ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 1987  Vcvv 3190  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-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
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-rex 2914  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-sn 4156  df-pr 4158  df-uni 4410  df-suc 5698
This theorem is referenced by:  orduninsuc  7005  tfindsg  7022  tfinds2  7025  finds  7054  findsg  7055  finds2  7056  seqomlem1  7505  oasuc  7564  onasuc  7568  infensuc  8098  phplem4  8102  php  8104  inf0  8478  inf3lem1  8485  dfom3  8504  cantnflt  8529  cantnflem1  8546  cnfcom  8557  infxpenlem  8796  pwsdompw  8986  ackbij1lem5  9006  cfslb2n  9050  cfsmolem  9052  fin1a2lem12  9193  axdc4lem  9237  alephreg  9364  bnj986  30785  bnj1018  30793  dfon2lem7  31448  bj-1ex  32638  bj-2ex  32639  dford3lem2  37113
  Copyright terms: Public domain W3C validator