Theorem unss12 3335

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 3332	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶))
2		unss2 3334	. 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐵 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷))
3	1, 2	sylan9ss 3196	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ∪ cun 3155   ⊆ wss 3157

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170

This theorem is referenced by:  ifssun  3575  fun  5430  resasplitss  5437  lspun  13958