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Theorem trsuc 4109
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 4101 . . . . . 6 B ⊆ suc B
2 ssexg 3870 . . . . . 6 ((B ⊆ suc B suc B A) → B V)
31, 2mpan 402 . . . . 5 (suc B AB V)
4 sucidg 4102 . . . . 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 3835 . . 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 1374  Vcvv 2535  wss 2894  Tr wtr 3828  suc csuc 4051
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-uni 3555  df-tr 3829  df-suc 4057
This theorem is referenced by: (None)
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