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Theorem sucssel 6262
 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 6248 . 2 (𝐴𝑉𝐴 ∈ suc 𝐴)
2 ssel 3886 . 2 (suc 𝐴𝐵 → (𝐴 ∈ suc 𝐴𝐴𝐵))
31, 2syl5com 31 1 (𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2112   ⊆ wss 3859  suc csuc 6172 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-un 3864  df-in 3866  df-ss 3876  df-sn 4524  df-suc 6176 This theorem is referenced by:  suc11  6273  ordelsuc  7535  ordsucelsuc  7537  oaordi  8183  nnaordi  8255  unbnn2  8801  ackbij1b  9692  ackbij2  9696  cflm  9703  isf32lem2  9807  indpi  10360  dfon2lem3  33270
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