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Theorem trsuc 4187
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 4180 . . . . . 6 𝐵 ⊆ suc 𝐵
2 ssexg 3924 . . . . . 6 ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵 ∈ V)
31, 2mpan 408 . . . . 5 (suc 𝐵𝐴𝐵 ∈ V)
4 sucidg 4181 . . . . 5 (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
53, 4syl 14 . . . 4 (suc 𝐵𝐴𝐵 ∈ suc 𝐵)
65ancri 311 . . 3 (suc 𝐵𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴))
7 trel 3889 . . 3 (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵𝐴))
86, 7syl5 32 . 2 (Tr 𝐴 → (suc 𝐵𝐴𝐵𝐴))
98imp 119 1 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wcel 1409  Vcvv 2574  wss 2945  Tr wtr 3882  suc csuc 4130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-uni 3609  df-tr 3883  df-suc 4136
This theorem is referenced by: (None)
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