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Theorem sucssel 4346
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 4338 . 2 (𝐴𝑉𝐴 ∈ suc 𝐴)
2 ssel 3091 . 2 (suc 𝐴𝐵 → (𝐴 ∈ suc 𝐴𝐴𝐵))
31, 2syl5com 29 1 (𝐴𝑉 → (suc 𝐴𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1480  wss 3071  suc csuc 4287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-suc 4293
This theorem is referenced by:  ordelsuc  4421  bj-nnelirr  13181
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