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Theorem sucssel 6282
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 6268 . 2 (𝐴𝑉𝐴 ∈ suc 𝐴)
2 ssel 3960 . 2 (suc 𝐴𝐵 → (𝐴 ∈ suc 𝐴𝐴𝐵))
31, 2syl5com 31 1 (𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2110  wss 3935  suc csuc 6192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-un 3940  df-in 3942  df-ss 3951  df-sn 4567  df-suc 6196
This theorem is referenced by:  suc11  6293  ordelsuc  7534  ordsucelsuc  7536  oaordi  8171  nnaordi  8243  unbnn2  8774  ackbij1b  9660  ackbij2  9664  cflm  9671  isf32lem2  9775  indpi  10328  dfon2lem3  33030
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