Theorem trsuc 4424

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 4417	. . . . . 6 ⊢ 𝐵 ⊆ suc 𝐵
2		ssexg 4144	. . . . . 6 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V)
3	1, 2	mpan 424	. . . . 5 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V)
4		sucidg 4418	. . . . 5 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵)
5	3, 4	syl 14	. . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ suc 𝐵)
6	5	ancri 324	. . 3 ⊢ (suc 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴))
7		trel 4110	. . 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 124	1 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2148  Vcvv 2739   ⊆ wss 3131  Tr wtr 4103  suc csuc 4367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4123

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-uni 3812  df-tr 4104  df-suc 4373

This theorem is referenced by:  nnnninf  7126