Theorem ssun 4161

Description: A condition that implies inclusion in the union of two classes. (Contributed by NM, 23-Nov-2003.)

Assertion

Ref	Expression
ssun	⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶))

Proof of Theorem ssun

Step	Hyp	Ref	Expression
1		ssun3 4146	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶))
2		ssun4 4147	. 2 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝐵 ∪ 𝐶))
3	1, 2	jaoi 857	1 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   ∨ wo 847   ∪ cun 3915   ⊆ wss 3917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-un 3922  df-ss 3934

This theorem is referenced by:  pwssun  5533  ordssun  6439  padct  32650