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Theorem trsuc 4314
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 4307 . . . . . 6 𝐵 ⊆ suc 𝐵
2 ssexg 4037 . . . . . 6 ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵 ∈ V)
31, 2mpan 420 . . . . 5 (suc 𝐵𝐴𝐵 ∈ V)
4 sucidg 4308 . . . . 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 4003 . . 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 1465  Vcvv 2660  wss 3041  Tr wtr 3996  suc csuc 4257
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-uni 3707  df-tr 3997  df-suc 4263
This theorem is referenced by:  nnnninf  6991
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