Theorem trsuc 6454

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 5228	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴))
2		sssucid 6447	. . . . 5 ⊢ 𝐵 ⊆ suc 𝐵
3		ssexg 5292	. . . . 5 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V)
4	2, 3	mpan 703	. . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V)
5		sucidg 6448	. . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
6	4, 5	syl 18	. . 3 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ suc 𝐵)
7	6	ancri 559	. 2 ⊢ (suc 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴))
8	1, 7	impel 515	1 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2146  Vcvv 3457   ⊆ wss 3906  Tr wtr 5220  suc csuc 6366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-un 3911  df-in 3913  df-ss 3923  df-sn 4592  df-uni 4875  df-tr 5221  df-suc 6370

This theorem is used by:  onuninsuci  7842  limsuc  7851  tz7.44-2  8400  cantnflt  9648  cantnfp1lem3  9656  cantnflem1b  9662  cantnflem1  9665  cnfcom  9676  axdc3lem2  10450  inar1  10775  bnj967  35398  limsuc2  43826