Theorem ssun3 4133

Description: Subclass law for union of classes. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
ssun3	⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶))

Proof of Theorem ssun3

Step	Hyp	Ref	Expression
1		ssun1 4131	. 2 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
2		sstr2 3945	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (𝐵 ∪ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶)))
3	1, 2	mpi 21	1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶))

This proof depends on syntax axioms:   → wi 4   ∪ cun 3904   ⊆ wss 3906

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 2148  ax-9 2156  ax-ext 2737

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 2744  df-cleq 2757  df-clel 2840  df-v 3459  df-un 3911  df-ss 3923

This theorem is used by:  ssun  4148  ssunsn2  4795  xpsspw  5798  uncmp  23612  alexsubALTlem3  24259  constrextdg2lem  34204  sxbrsigalem0  34728  bnj1450  35505  fineqvac  35588  altxpsspw  36508  ttcuniun  37080  ttciunun  37081  pibt2  38122  superuncl  44354  cnvtrcl0  44412