Theorem trsucss 4563

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 4543	. 2 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
2		trss 4233	. . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
3		eqimss 3302	. . . 4 ⊢ (𝐵 = 𝐴 → 𝐵 ⊆ 𝐴)
4	3	a1i 9	. . 3 ⊢ (Tr 𝐴 → (𝐵 = 𝐴 → 𝐵 ⊆ 𝐴))
5	2, 4	jaod 729	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) → 𝐵 ⊆ 𝐴))
6	1, 5	syl5 32	1 ⊢ (Tr 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐵 ⊆ 𝐴))

Syntax hints:   → wi 4   ∨ wo 720   = wceq 1402   ∈ wcel 2209   ⊆ 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

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-ral 2533  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:  onsucsssucr  4651  ordpwsucss  4709  nnnninfeq  7458  bj-el2oss1o  16716  nnsf  16953