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Theorem trsuc 4125
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 B A) → B A)

Proof of Theorem trsuc
StepHypRef Expression
1 sssucid 4118 . . . . . 6 B ⊆ suc B
2 ssexg 3887 . . . . . 6 ((B ⊆ suc B suc B A) → B V)
31, 2mpan 400 . . . . 5 (suc B AB V)
4 sucidg 4119 . . . . 5 (B V → B suc B)
53, 4syl 14 . . . 4 (suc B AB suc B)
65ancri 307 . . 3 (suc B A → (B suc B suc B A))
7 trel 3852 . . 3 (Tr A → ((B suc B suc B A) → B A))
86, 7syl5 28 . 2 (Tr A → (suc B AB A))
98imp 115 1 ((Tr A suc B A) → B A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wcel 1390  Vcvv 2551  wss 2911  Tr wtr 3845  suc csuc 4068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-uni 3572  df-tr 3846  df-suc 4074
This theorem is referenced by: (None)
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