Theorem ssun4 4161

Description: Subclass law for union of classes. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
ssun4	⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵))

Proof of Theorem ssun4

Step	Hyp	Ref	Expression
1		ssun2 4159	. 2 ⊢ 𝐵 ⊆ (𝐶 ∪ 𝐵)
2		sstr2 3970	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (𝐶 ∪ 𝐵) → 𝐴 ⊆ (𝐶 ∪ 𝐵)))
3	1, 2	mpi 20	1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵))

Syntax hints:   → wi 4   ∪ cun 3929   ⊆ wss 3931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936  df-ss 3948

This theorem is referenced by:  ssun  4175  xpsspw  5793  uncmp  23346  volcn  25564  bnj1408  35072  bnj1452  35088  pibt2  37440  elrfi  42684  cnvrcl0  43616