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Theorem trsuc 6272
 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 5175 . 2 (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵𝐴))
2 sssucid 6265 . . . . 5 𝐵 ⊆ suc 𝐵
3 ssexg 5223 . . . . 5 ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵 ∈ V)
42, 3mpan 686 . . . 4 (suc 𝐵𝐴𝐵 ∈ V)
5 sucidg 6266 . . . 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 2107  Vcvv 3499   ⊆ wss 3939  Tr wtr 5168  suc csuc 6190 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-rab 3151  df-v 3501  df-un 3944  df-in 3946  df-ss 3955  df-sn 4564  df-uni 4837  df-tr 5169  df-suc 6194 This theorem is referenced by:  onuninsuci  7546  limsuc  7555  tz7.44-2  8037  cantnflt  9127  cantnfp1lem3  9135  cantnflem1b  9141  cantnflem1  9144  cnfcom  9155  axdc3lem2  9865  inar1  10189  bnj967  32105  limsuc2  39508
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