Theorem ssun3 4111

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 4109	. 2 ⊢ 𝐵 ⊆ (𝐵 ∪ 𝐶)
2		sstr2 3924	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ (𝐵 ∪ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶)))
3	1, 2	mpi 20	1 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   ∪ cun 3883   ⊆ wss 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-v 3429  df-un 3890  df-ss 3902

This theorem is referenced by:  ssun  4126  ssunsn2  4760  xpsspw  5754  uncmp  23356  alexsubALTlem3  24002  constrextdg2lem  33880  sxbrsigalem0  34403  bnj1450  35180  fineqvac  35248  altxpsspw  36147  ttcuniun  36680  ttciunun  36681  pibt2  37721  superuncl  43983  cnvtrcl0  44041