Theorem ssun4 4127

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 4125	. 2 ⊢ 𝐵 ⊆ (𝐶 ∪ 𝐵)
2		sstr2 3938	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (𝐶 ∪ 𝐵) → 𝐴 ⊆ (𝐶 ∪ 𝐵)))
3	1, 2	mpi 21	1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐶 ∪ 𝐵))

This proof depends on syntax axioms:   → wi 4   ∪ cun 3897   ⊆ wss 3899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3904  df-ss 3916

This theorem is used by:  ssun  4141  xpsspw  5790  uncmp  23628  volcn  25834  bnj1408  35545  bnj1452  35561  tz9.1regs  35660  pibt2  38171  elrfi  43539  cnvrcl0  44465