Theorem ssun 3443

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 3429	. 2 ⊢ (A ⊆ B → A ⊆ (B ∪ C))
2		ssun4 3430	. 2 ⊢ (A ⊆ C → A ⊆ (B ∪ C))
3	1, 2	jaoi 368	1 ⊢ ((A ⊆ B ∨ A ⊆ C) → A ⊆ (B ∪ C))

Syntax hints:   → wi 4   ∨ wo 357   ∪ cun 3208   ⊆ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-ss 3260

This theorem is referenced by: (None)