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Theorem trsucss 5780
 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 5760 . 2 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
2 trss 4731 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
3 eqimss 3642 . . . 4 (𝐵 = 𝐴𝐵𝐴)
43a1i 11 . . 3 (Tr 𝐴 → (𝐵 = 𝐴𝐵𝐴))
52, 4jaod 395 . 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 383   = wceq 1480   ∈ wcel 1987   ⊆ wss 3560  Tr wtr 4722  suc csuc 5694 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-v 3192  df-un 3565  df-in 3567  df-ss 3574  df-sn 4156  df-uni 4410  df-tr 4723  df-suc 5698 This theorem is referenced by:  efgmnvl  18067
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