Theorem ssun3 3292

Description: Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ssun3	⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶))

Proof of Theorem ssun3

Step	Hyp	Ref	Expression
1		ssun1 3290	. 2 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
2		sstr2 3154	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (𝐵 ∪ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶)))
3	1, 2	mpi 15	1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   ∪ cun 3119   ⊆ wss 3121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134

This theorem is referenced by:  ssun  3306  xpsspw  4723