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Theorem trsuc 6268
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 5170 . 2 (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵𝐴))
2 sssucid 6261 . . . . 5 𝐵 ⊆ suc 𝐵
3 ssexg 5218 . . . . 5 ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵𝐴) → 𝐵 ∈ V)
42, 3mpan 686 . . . 4 (suc 𝐵𝐴𝐵 ∈ V)
5 sucidg 6262 . . . 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 2105  Vcvv 3492  wss 3933  Tr wtr 5163  suc csuc 6186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-un 3938  df-in 3940  df-ss 3949  df-sn 4558  df-uni 4831  df-tr 5164  df-suc 6190
This theorem is referenced by:  onuninsuci  7544  limsuc  7553  tz7.44-2  8032  cantnflt  9123  cantnfp1lem3  9131  cantnflem1b  9137  cantnflem1  9140  cnfcom  9151  axdc3lem2  9861  inar1  10185  bnj967  32116  limsuc2  39519
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