Theorem ssun4 4142

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 4140	. 2 ⊢ 𝐵 ⊆ (𝐶 ∪ 𝐵)
2		sstr2 3952	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (𝐶 ∪ 𝐵) → 𝐴 ⊆ (𝐶 ∪ 𝐵)))
3	1, 2	mpi 21	1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵))

Syntax hints:   → wi 4   ∪ cun 3911   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-un 3918  df-ss 3930

This theorem is referenced by:  ssun  4156  xpsspw  5794  uncmp  23525  volcn  25730  bnj1408  35365  bnj1452  35381  tz9.1regs  35466  pibt2  37946  elrfi  43312  cnvrcl0  44238