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Theorem trsucss 4126
 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 4106 . 2 (B suc A → (B A B = A))
2 trss 3854 . . 3 (Tr A → (B ABA))
3 eqimss 2991 . . . 4 (B = ABA)
43a1i 9 . . 3 (Tr A → (B = ABA))
52, 4jaod 636 . 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 628   = wceq 1242   ∈ wcel 1390   ⊆ wss 2911  Tr wtr 3845  suc csuc 4068 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-uni 3572  df-tr 3846  df-suc 4074 This theorem is referenced by:  onsucsssucr  4200  ordpwsucss  4243
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