Theorem trsuc 6482

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 5292	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴))
2		sssucid 6475	. . . . 5 ⊢ 𝐵 ⊆ suc 𝐵
3		ssexg 5341	. . . . 5 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V)
4	2, 3	mpan 689	. . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V)
5		sucidg 6476	. . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
6	4, 5	syl 17	. . 3 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ suc 𝐵)
7	6	ancri 549	. 2 ⊢ (suc 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴))
8	1, 7	impel 505	1 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2108  Vcvv 3488   ⊆ wss 3976  Tr wtr 5283  suc csuc 6397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-un 3981  df-in 3983  df-ss 3993  df-sn 4649  df-uni 4932  df-tr 5284  df-suc 6401

This theorem is referenced by:  onuninsuci  7877  limsuc  7886  tz7.44-2  8463  cantnflt  9741  cantnfp1lem3  9749  cantnflem1b  9755  cantnflem1  9758  cnfcom  9769  axdc3lem2  10520  inar1  10844  bnj967  34921  limsuc2  42998