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

Theorem sucssel 6449
Description: A set whose successor is a subset of another class is a member of that class. (Contributed by NM, 16-Sep-1995.)
Assertion
Ref Expression
sucssel (𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))

Proof of Theorem sucssel
StepHypRef Expression
1 sucidg 6435 . 2 (𝐴𝑉𝐴 ∈ suc 𝐴)
2 ssel 3967 . 2 (suc 𝐴𝐵 → (𝐴 ∈ suc 𝐴𝐴𝐵))
31, 2syl5com 31 1 (𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wss 3940  suc csuc 6356
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3945  df-in 3947  df-ss 3957  df-sn 4621  df-suc 6360
This theorem is referenced by:  suc11  6461  ordelsuc  7801  ordsucelsuc  7803  oaordi  8541  nnaordi  8613  unbnn2  9295  ackbij1b  10229  ackbij2  10233  cflm  10240  isf32lem2  10344  indpi  10897  dfon2lem3  35218
  Copyright terms: Public domain W3C validator