Theorem trsuc 6447

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 5220	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴))
2		sssucid 6440	. . . . 5 ⊢ 𝐵 ⊆ suc 𝐵
3		ssexg 5284	. . . . 5 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V)
4	2, 3	mpan 703	. . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V)
5		sucidg 6441	. . . 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 2145  Vcvv 3450   ⊆ wss 3899  Tr wtr 5212  suc csuc 6359

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 2147  ax-9 2155  ax-ext 2732  ax-sep 5251

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-un 3904  df-in 3906  df-ss 3916  df-sn 4585  df-uni 4868  df-tr 5213  df-suc 6363

This theorem is used by:  onuninsuci  7837  limsuc  7846  tz7.44-2  8397  cantnflt  9652  cantnfp1lem3  9660  cantnflem1b  9666  cantnflem1  9669  cnfcom  9680  axdc3lem2  10454  inar1  10785  bnj967  35455  limsuc2  43883