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Theorem trsuc 4304
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 4297 . . . . . 6 𝐵 ⊆ suc 𝐵
2 ssexg 4027 . . . . . 6 ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵 ∈ V)
31, 2mpan 418 . . . . 5 (suc 𝐵𝐴𝐵 ∈ V)
4 sucidg 4298 . . . . 5 (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
53, 4syl 14 . . . 4 (suc 𝐵𝐴𝐵 ∈ suc 𝐵)
65ancri 320 . . 3 (suc 𝐵𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴))
7 trel 3993 . . 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 1463  Vcvv 2657  wss 3037  Tr wtr 3986  suc csuc 4247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-sn 3499  df-uni 3703  df-tr 3987  df-suc 4253
This theorem is referenced by:  nnnninf  6973
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