Theorem unss12 3436

Description: Subclass law for union of classes. (Contributed by NM, 2-Jun-2004.)

Assertion

Ref	Expression
unss12	⊢ ((A ⊆ B ∧ C ⊆ D) → (A ∪ C) ⊆ (B ∪ D))

Proof of Theorem unss12

Step	Hyp	Ref	Expression
1		unss1 3433	. 2 ⊢ (A ⊆ B → (A ∪ C) ⊆ (B ∪ C))
2		unss2 3435	. 2 ⊢ (C ⊆ D → (B ∪ C) ⊆ (B ∪ D))
3	1, 2	sylan9ss 3286	1 ⊢ ((A ⊆ B ∧ C ⊆ D) → (A ∪ C) ⊆ (B ∪ D))

Syntax hints:   → wi 4   ∧ wa 358   ∪ cun 3208   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260

This theorem is referenced by:  fun  5237