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

Theorem sucssel 6479
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 6465 . 2 (𝐴𝑉𝐴 ∈ suc 𝐴)
2 ssel 3977 . 2 (suc 𝐴𝐵 → (𝐴 ∈ suc 𝐴𝐴𝐵))
31, 2syl5com 31 1 (𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wss 3951  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-sn 4627  df-suc 6390
This theorem is referenced by:  suc11  6491  ordelsuc  7840  ordsucelsuc  7842  oaordi  8584  nnaordi  8656  unbnn2  9333  ackbij1b  10278  ackbij2  10282  cflm  10290  isf32lem2  10394  indpi  10947  dfon2lem3  35786
  Copyright terms: Public domain W3C validator