Theorem ssun 3260

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 3246	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ (𝐵 ∪ 𝐶))
2		ssun4 3247	. 2 ⊢ (𝐴 ⊆ 𝐶 → 𝐴 ⊆ (𝐵 ∪ 𝐶))
3	1, 2	jaoi 706	1 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ (𝐵 ∪ 𝐶))

Syntax hints:   → wi 4   ∨ wo 698   ∪ cun 3074   ⊆ wss 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089

This theorem is referenced by:  pwunss  4213  pwssunim  4214