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Theorem trsucss 6027
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 6008 . 2 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
2 trss 4955 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
3 eqimss 3854 . . . 4 (𝐵 = 𝐴𝐵𝐴)
43a1i 11 . . 3 (Tr 𝐴 → (𝐵 = 𝐴𝐵𝐴))
52, 4jaod 886 . 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 874   = wceq 1653  wcel 2157  wss 3770  Tr wtr 4946  suc csuc 5944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2778
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-v 3388  df-un 3775  df-in 3777  df-ss 3784  df-sn 4370  df-uni 4630  df-tr 4947  df-suc 5948
This theorem is referenced by:  efgmnvl  18439
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