Theorem ssun 1634

Description: A condition that implies inclusion in the union of two classes.

Assertion

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

Proof of Theorem ssun

Step	Hyp	Ref	Expression
1		ssun3 1623	. 2 ⊢ (A ⊆ B → A ⊆ (B ∪ C))
2		ssun4 1624	. 2 ⊢ (A ⊆ C → A ⊆ (B ∪ C))
3	1, 2	jaoi 275	1 ⊢ ((A ⊆ B ∨ A ⊆ C) → A ⊆ (B ∪ C))

Syntax hints:   → wi 2   ∨ wo 195   ∪ cun 1485   ⊆ wss 1487

This theorem is referenced by:  pwunss 1916  pwssun 1917  ordssun 2330

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-ss 1492

metamath.org