Theorem trsuc 4125

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		sssucid 4118	. . . . . 6 ⊢ B ⊆ suc B
2		ssexg 3887	. . . . . 6 ⊢ ((B ⊆ suc B ∧ suc B ∈ A) → B ∈ V)
3	1, 2	mpan 400	. . . . 5 ⊢ (suc B ∈ A → B ∈ V)
4		sucidg 4119	. . . . 5 ⊢ (B ∈ V → B ∈ suc B)
5	3, 4	syl 14	. . . 4 ⊢ (suc B ∈ A → B ∈ suc B)
6	5	ancri 307	. . 3 ⊢ (suc B ∈ A → (B ∈ suc B ∧ suc B ∈ A))
7		trel 3852	. . 3 ⊢ (Tr A → ((B ∈ suc B ∧ suc B ∈ A) → B ∈ A))
8	6, 7	syl5 28	. 2 ⊢ (Tr A → (suc B ∈ A → B ∈ A))
9	8	imp 115	1 ⊢ ((Tr A ∧ suc B ∈ A) → B ∈ A)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911  Tr wtr 3845  suc csuc 4068

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-uni 3572  df-tr 3846  df-suc 4074

This theorem is referenced by: (None)