Theorem trsuc 6258

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 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)

Proof of Theorem trsuc

Step	Hyp	Ref	Expression
1		trel 5149	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴))
2		sssucid 6251	. . . . 5 ⊢ 𝐵 ⊆ suc 𝐵
3		ssexg 5197	. . . . 5 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V)
4	2, 3	mpan 689	. . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V)
5		sucidg 6252	. . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
6	4, 5	syl 17	. . 3 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ suc 𝐵)
7	6	ancri 553	. 2 ⊢ (suc 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴))
8	1, 7	impel 509	1 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3860  Tr wtr 5142  suc csuc 6176

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5173

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-rab 3079  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-uni 4802  df-tr 5143  df-suc 6180

This theorem is referenced by:  onuninsuci  7560  limsuc  7569  tz7.44-2  8059  cantnflt  9181  cantnfp1lem3  9189  cantnflem1b  9195  cantnflem1  9198  cnfcom  9209  axdc3lem2  9924  inar1  10248  bnj967  32457  limsuc2  40393