Theorem intssuni2 3951

Description: Subclass relationship for intersection and union. (Contributed by NM, 29-Jul-2006.)

Assertion

Ref	Expression
intssuni2	⊢ ((A ⊆ B ∧ A ≠ ∅) → ∩A ⊆ ∪B)

Proof of Theorem intssuni2

Step	Hyp	Ref	Expression
1		intssuni 3948	. 2 ⊢ (A ≠ ∅ → ∩A ⊆ ∪A)
2		uniss 3912	. 2 ⊢ (A ⊆ B → ∪A ⊆ ∪B)
3	1, 2	sylan9ssr 3286	1 ⊢ ((A ⊆ B ∧ A ≠ ∅) → ∩A ⊆ ∪B)

Syntax hints:   → wi 4   ∧ wa 358   ≠ wne 2516   ⊆ wss 3257  ∅c0 3550  ∪cuni 3891  ∩cint 3926

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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-uni 3892  df-int 3927

This theorem is referenced by:  rintn0  4056