Theorem trsuc 4109

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 4101	. . . . . 6 ⊢ B ⊆ suc B
2		ssexg 3870	. . . . . 6 ⊢ ((B ⊆ suc B ∧ suc B ∈ A) → B ∈ V)
3	1, 2	mpan 402	. . . . 5 ⊢ (suc B ∈ A → B ∈ V)
4		sucidg 4102	. . . . 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 3835	. . 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 1374  Vcvv 2535   ⊆ wss 2894  Tr wtr 3828  suc csuc 4051

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-sn 3356  df-uni 3555  df-tr 3829  df-suc 4057

This theorem is referenced by: (None)