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

Theorem sucexg 6595
Description: The successor of a set is a set (generalization). (Contributed by NM, 5-Jun-1994.)
Assertion
Ref Expression
sucexg  |-  ( A  e.  V  ->  suc  A  e.  _V )

Proof of Theorem sucexg
StepHypRef Expression
1 elex 3031 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 sucexb 6594 . 2  |-  ( A  e.  _V  <->  suc  A  e. 
_V )
31, 2sylib 199 1  |-  ( A  e.  V  ->  suc  A  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1872   _Vcvv 3022   suc csuc 5387
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-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541
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 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-rex 2720  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-sn 3942  df-pr 3944  df-uni 4163  df-suc 5391
This theorem is referenced by:  sucex  6596  suceloni  6598  hsmexlem1  8807  dfon2lem3  30382
  Copyright terms: Public domain W3C validator