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Theorem trsuc 4381
 Description: A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
trsuc ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)

Proof of Theorem trsuc
StepHypRef Expression
1 sssucid 4374 . . . . . 6 𝐵 ⊆ suc 𝐵
2 ssexg 4103 . . . . . 6 ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵 ∈ V)
31, 2mpan 421 . . . . 5 (suc 𝐵𝐴𝐵 ∈ V)
4 sucidg 4375 . . . . 5 (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
53, 4syl 14 . . . 4 (suc 𝐵𝐴𝐵 ∈ suc 𝐵)
65ancri 322 . . 3 (suc 𝐵𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴))
7 trel 4069 . . 3 (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵𝐴))
86, 7syl5 32 . 2 (Tr 𝐴 → (suc 𝐵𝐴𝐵𝐴))
98imp 123 1 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 2128  Vcvv 2712   ⊆ wss 3102  Tr wtr 4062  suc csuc 4324 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-sep 4082 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-sn 3566  df-uni 3773  df-tr 4063  df-suc 4330 This theorem is referenced by:  nnnninf  7058
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