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Theorem trsucss 4214
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 4194 . 2 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
2 trss 3910 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
3 eqimss 3062 . . . 4 (𝐵 = 𝐴𝐵𝐴)
43a1i 9 . . 3 (Tr 𝐴 → (𝐵 = 𝐴𝐵𝐴))
52, 4jaod 670 . 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 662   = wceq 1285  wcel 1434  wss 2984  Tr wtr 3901  suc csuc 4156
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-sn 3428  df-uni 3628  df-tr 3902  df-suc 4162
This theorem is referenced by:  onsucsssucr  4289  ordpwsucss  4346
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