Theorem ssun4 4126

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 4124	. 2 ⊢ 𝐵 ⊆ (𝐶 ∪ 𝐵)
2		sstr2 3936	. 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 3895   ⊆ wss 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-un 3902  df-ss 3914

This theorem is referenced by:  ssun  4140  xpsspw  5744  uncmp  23313  volcn  25529  bnj1408  35040  bnj1452  35056  tz9.1regs  35122  pibt2  37451  elrfi  42727  cnvrcl0  43658