Theorem trsuc 4562

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 4555	. . . . . 6 ⊢ 𝐵 ⊆ suc 𝐵
2		ssexg 4267	. . . . . 6 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V)
3	1, 2	mpan 428	. . . . 5 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V)
4		sucidg 4556	. . . . 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 4231	. . 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 2209  Vcvv 2821   ⊆ wss 3220  Tr wtr 4224  suc csuc 4505

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-sep 4244

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3711  df-uni 3931  df-tr 4225  df-suc 4511

This theorem is referenced by:  nnnninf  7456