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Theorem sucssel 6033
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 6019 . 2 (𝐴𝑉𝐴 ∈ suc 𝐴)
2 ssel 3792 . 2 (suc 𝐴𝐵 → (𝐴 ∈ suc 𝐴𝐴𝐵))
31, 2syl5com 31 1 (𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2157  wss 3769  suc csuc 5943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-un 3774  df-in 3776  df-ss 3783  df-sn 4369  df-suc 5947
This theorem is referenced by:  suc11  6044  ordelsuc  7254  ordsucelsuc  7256  oaordi  7866  nnaordi  7938  unbnn2  8459  ackbij1b  9349  ackbij2  9353  cflm  9360  isf32lem2  9464  indpi  10017  dfon2lem3  32202
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