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Theorem sucssel 5857
 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 5841 . 2 (𝐴𝑉𝐴 ∈ suc 𝐴)
2 ssel 3630 . 2 (suc 𝐴𝐵 → (𝐴 ∈ suc 𝐴𝐴𝐵))
31, 2syl5com 31 1 (𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2030   ⊆ wss 3607  suc csuc 5763 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-in 3614  df-ss 3621  df-sn 4211  df-suc 5767 This theorem is referenced by:  suc11  5869  ordelsuc  7062  ordsucelsuc  7064  oaordi  7671  nnaordi  7743  unbnn2  8258  ackbij1b  9099  ackbij2  9103  cflm  9110  isf32lem2  9214  indpi  9767  dfon2lem3  31814
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