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Theorem trsucss 4345
Description: A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)
Assertion
Ref Expression
trsucss (Tr 𝐴 → (𝐵 ∈ suc 𝐴𝐵𝐴))

Proof of Theorem trsucss
StepHypRef Expression
1 elsuci 4325 . 2 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
2 trss 4035 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
3 eqimss 3151 . . . 4 (𝐵 = 𝐴𝐵𝐴)
43a1i 9 . . 3 (Tr 𝐴 → (𝐵 = 𝐴𝐵𝐴))
52, 4jaod 706 . 2 (Tr 𝐴 → ((𝐵𝐴𝐵 = 𝐴) → 𝐵𝐴))
61, 5syl5 32 1 (Tr 𝐴 → (𝐵 ∈ suc 𝐴𝐵𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wo 697   = wceq 1331  wcel 1480  wss 3071  Tr wtr 4026  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-ral 2421  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-uni 3737  df-tr 4027  df-suc 4293
This theorem is referenced by:  onsucsssucr  4425  ordpwsucss  4482  bj-el2oss1o  13040  nnsf  13260  nninfalllemn  13263
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