Theorem trsuc 6452

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 5275	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴))
2		sssucid 6445	. . . . 5 ⊢ 𝐵 ⊆ suc 𝐵
3		ssexg 5324	. . . . 5 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V)
4	2, 3	mpan 689	. . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V)
5		sucidg 6446	. . . 4 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
6	4, 5	syl 17	. . 3 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ suc 𝐵)
7	6	ancri 551	. 2 ⊢ (suc 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴))
8	1, 7	impel 507	1 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107  Vcvv 3475   ⊆ wss 3949  Tr wtr 5266  suc csuc 6367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-sn 4630  df-uni 4910  df-tr 5267  df-suc 6371

This theorem is referenced by:  onuninsuci  7829  limsuc  7838  tz7.44-2  8407  cantnflt  9667  cantnfp1lem3  9675  cantnflem1b  9681  cantnflem1  9684  cnfcom  9695  axdc3lem2  10446  inar1  10770  bnj967  33956  limsuc2  41783