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Theorem trsuc 6440
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 5267 . 2 (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵𝐴))
2 sssucid 6433 . . . . 5 𝐵 ⊆ suc 𝐵
3 ssexg 5316 . . . . 5 ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵 ∈ V)
42, 3mpan 688 . . . 4 (suc 𝐵𝐴𝐵 ∈ V)
5 sucidg 6434 . . . 4 (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
64, 5syl 17 . . 3 (suc 𝐵𝐴𝐵 ∈ suc 𝐵)
76ancri 550 . 2 (suc 𝐵𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴))
81, 7impel 506 1 ((Tr 𝐴 ∧ suc 𝐵𝐴) → 𝐵𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2106  Vcvv 3473  wss 3944  Tr wtr 5258  suc csuc 6355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702  ax-sep 5292
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-rab 3432  df-v 3475  df-un 3949  df-in 3951  df-ss 3961  df-sn 4623  df-uni 4902  df-tr 5259  df-suc 6359
This theorem is referenced by:  onuninsuci  7812  limsuc  7821  tz7.44-2  8389  cantnflt  9649  cantnfp1lem3  9657  cantnflem1b  9663  cantnflem1  9666  cnfcom  9677  axdc3lem2  10428  inar1  10752  bnj967  33787  limsuc2  41554
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