Theorem unss12 3349

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 3346	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐶))
2		unss2 3348	. 2 ⊢ (𝐶 ⊆ 𝐷 → (𝐵 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷))
3	1, 2	sylan9ss 3210	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝐴 ∪ 𝐶) ⊆ (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ∪ cun 3168   ⊆ wss 3170

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-in 3176  df-ss 3183

This theorem is referenced by:  ifssun  3589  fun  5458  resasplitss  5466  lspun  14234