Theorem trsuc 4407

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 4400	. . . . . 6 ⊢ 𝐵 ⊆ suc 𝐵
2		ssexg 4128	. . . . . 6 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V)
3	1, 2	mpan 422	. . . . 5 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V)
4		sucidg 4401	. . . . 5 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
5	3, 4	syl 14	. . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ suc 𝐵)
6	5	ancri 322	. . 3 ⊢ (suc 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴))
7		trel 4094	. . 3 ⊢ (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴))
8	6, 7	syl5 32	. 2 ⊢ (Tr 𝐴 → (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴))
9	8	imp 123	1 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   ∈ wcel 2141  Vcvv 2730   ⊆ wss 3121  Tr wtr 4087  suc csuc 4350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-uni 3797  df-tr 4088  df-suc 4356

This theorem is referenced by:  nnnninf  7102