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Theorem trsuc 6452
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 trel 5275 . 2 (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵𝐴))
2 sssucid 6445 . . . . 5 𝐵 ⊆ suc 𝐵
3 ssexg 5324 . . . . 5 ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵 ∈ V)
42, 3mpan 686 . . . 4 (suc 𝐵𝐴𝐵 ∈ V)
5 sucidg 6446 . . . 4 (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
64, 5syl 17 . . 3 (suc 𝐵𝐴𝐵 ∈ suc 𝐵)
76ancri 548 . 2 (suc 𝐵𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴))
81, 7impel 504 1 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  wcel 2104  Vcvv 3472  wss 3949  Tr wtr 5266  suc csuc 6367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-un 3954  df-in 3956  df-ss 3966  df-sn 4630  df-uni 4910  df-tr 5267  df-suc 6371
This theorem is referenced by:  onuninsuci  7833  limsuc  7842  tz7.44-2  8411  cantnflt  9671  cantnfp1lem3  9679  cantnflem1b  9685  cantnflem1  9688  cnfcom  9699  axdc3lem2  10450  inar1  10774  bnj967  34252  limsuc2  42087
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