Theorem ssun4 4105

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 4103	. 2 ⊢ 𝐵 ⊆ (𝐶 ∪ 𝐵)
2		sstr2 3924	. 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 3881   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-in 3890  df-ss 3900

This theorem is referenced by:  ssun  4119  xpsspw  5708  dftrpred3g  9412  uncmp  22462  volcn  24675  bnj1408  32916  bnj1452  32932  pibt2  35515  elrfi  40432  cnvrcl0  41122