Theorem trsucss 6456

Description: A member of the successor of a transitive class is a subclass of it. Lemma 1.13 of [Schloeder] p. 2. (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 6435	. 2 ⊢ (𝐵 ∈ suc 𝐴 → (𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴))
2		trss 5221	. . 3 ⊢ (Tr 𝐴 → (𝐵 ∈ 𝐴 → 𝐵 ⊆ 𝐴))
3		eqimss 3967	. . . 4 ⊢ (𝐵 = 𝐴 → 𝐵 ⊆ 𝐴)
4	3	a1i 11	. . 3 ⊢ (Tr 𝐴 → (𝐵 = 𝐴 → 𝐵 ⊆ 𝐴))
5	2, 4	jaod 873	. 2 ⊢ (Tr 𝐴 → ((𝐵 ∈ 𝐴 ∨ 𝐵 = 𝐴) → 𝐵 ⊆ 𝐴))
6	1, 5	syl5 35	1 ⊢ (Tr 𝐴 → (𝐵 ∈ suc 𝐴 → 𝐵 ⊆ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ⊆ wss 3896  Tr wtr 5211  suc csuc 6367

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-ss 3897  df-un 4150  df-sn 4585  df-uni 4867  df-tr 5212  df-suc 6371

This theorem is used by:  efgmnvl  19937  ordsssucim  44541