Theorem unss12 3376

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 3373	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶))
2		unss2 3375	. 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐵 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷))
3	1, 2	sylan9ss 3237	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ∪ cun 3195   ⊆ wss 3197

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210

This theorem is referenced by:  ifssun  3617  fun  5493  resasplitss  5501  lspun  14351