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Theorem trsuc 5727
 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 4687 . 2 (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵𝐴))
2 sssucid 5719 . . . . 5 𝐵 ⊆ suc 𝐵
3 ssexg 4732 . . . . 5 ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵 ∈ V)
42, 3mpan 702 . . . 4 (suc 𝐵𝐴𝐵 ∈ V)
5 sucidg 5720 . . . 4 (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
64, 5syl 17 . . 3 (suc 𝐵𝐴𝐵 ∈ suc 𝐵)
76ancri 573 . 2 (suc 𝐵𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴))
81, 7impel 484 1 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  Tr wtr 4680  suc csuc 5642 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-uni 4373  df-tr 4681  df-suc 5646 This theorem is referenced by:  onuninsuci  6932  limsuc  6941  tz7.44-2  7390  cantnflt  8452  cantnfp1lem3  8460  cantnflem1b  8466  cantnflem1  8469  cnfcom  8480  axdc3lem2  9156  inar1  9476  bnj967  30269  limsuc2  36629
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