MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  trsucss Structured version   Visualization version   GIF version

Theorem trsucss 5953
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 5933 . 2 (𝐵 ∈ suc 𝐴 → (𝐵𝐴𝐵 = 𝐴))
2 trss 4896 . . 3 (Tr 𝐴 → (𝐵𝐴𝐵𝐴))
3 eqimss 3806 . . . 4 (𝐵 = 𝐴𝐵𝐴)
43a1i 11 . . 3 (Tr 𝐴 → (𝐵 = 𝐴𝐵𝐴))
52, 4jaod 848 . 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 836   = wceq 1631  wcel 2145  wss 3723  Tr wtr 4887  suc csuc 5867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-v 3353  df-un 3728  df-in 3730  df-ss 3737  df-sn 4318  df-uni 4576  df-tr 4888  df-suc 5871
This theorem is referenced by:  efgmnvl  18334
  Copyright terms: Public domain W3C validator