Theorem trsucss 4160

Description: A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.)

Assertion

Ref	Expression
trsucss	⊢ (Tr 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐵 ⊆ 𝐴))

Proof of Theorem trsucss

Step	Hyp	Ref	Expression
1		elsuci 4140	. 2 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
2		trss 3863	. . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
3		eqimss 2997	. . . 4 ⊢ (𝐵 = 𝐴 → 𝐵 ⊆ 𝐴)
4	3	a1i 9	. . 3 ⊢ (Tr 𝐴 → (𝐵 = 𝐴 → 𝐵 ⊆ 𝐴))
5	2, 4	jaod 637	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) → 𝐵 ⊆ 𝐴))
6	1, 5	syl5 28	1 ⊢ (Tr 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∨ wo 629   = wceq 1243   ∈ wcel 1393   ⊆ wss 2917  Tr wtr 3854  suc csuc 4102

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-uni 3581  df-tr 3855  df-suc 4108

This theorem is referenced by:  onsucsssucr  4235  ordpwsucss  4291