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Theorem sucssel 5807
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 5791 . 2 (𝐴𝑉𝐴 ∈ suc 𝐴)
2 ssel 3589 . 2 (suc 𝐴𝐵 → (𝐴 ∈ suc 𝐴𝐴𝐵))
31, 2syl5com 31 1 (𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1988  wss 3567  suc csuc 5713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-un 3572  df-in 3574  df-ss 3581  df-sn 4169  df-suc 5717
This theorem is referenced by:  suc11  5819  ordelsuc  7005  ordsucelsuc  7007  oaordi  7611  nnaordi  7683  unbnn2  8202  ackbij1b  9046  ackbij2  9050  cflm  9057  isf32lem2  9161  indpi  9714  dfon2lem3  31664
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