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Theorem trsucss 6254
 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 6235 . 2 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
2 trss 5147 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
3 eqimss 3948 . . . 4 (𝐵 = 𝐴𝐵𝐴)
43a1i 11 . . 3 (Tr 𝐴 → (𝐵 = 𝐴𝐵𝐴))
52, 4jaod 856 . 2 (Tr 𝐴 → ((𝐵𝐴𝐵 = 𝐴) → 𝐵𝐴))
61, 5syl5 34 1 (Tr 𝐴 → (𝐵 ∈ suc 𝐴𝐵𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ⊆ wss 3858  Tr wtr 5138  suc csuc 6171 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-v 3411  df-un 3863  df-in 3865  df-ss 3875  df-sn 4523  df-uni 4799  df-tr 5139  df-suc 6175 This theorem is referenced by:  efgmnvl  18907
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