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Theorem trsucss 4108
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 4087 . 2 (B suc A → (B A B = A))
2 trss 3836 . . 3 (Tr A → (B ABA))
3 eqimss 2973 . . . 4 (B = ABA)
43a1i 9 . . 3 (Tr A → (B = ABA))
52, 4jaod 624 . 2 (Tr A → ((B A B = A) → BA))
61, 5syl5 28 1 (Tr A → (B suc ABA))
Colors of variables: wff set class
Syntax hints:  wi 4   wo 616   = wceq 1228   wcel 1375  wss 2893  Tr wtr 3827  suc csuc 4049
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-ral 2288  df-v 2536  df-un 2898  df-in 2900  df-ss 2907  df-sn 3355  df-uni 3554  df-tr 3828  df-suc 4055
This theorem is referenced by:  onsucsssucr  4182  ordpwsucss  4225
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