Theorem intssuni2 3952

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 3949	. 2 ⊢ (A ≠ ∅ → ∩A ⊆ ∪A)
2		uniss 3913	. 2 ⊢ (A ⊆ B → ∪A ⊆ ∪B)
3	1, 2	sylan9ssr 3287	1 ⊢ ((A ⊆ B ∧ A ≠ ∅) → ∩A ⊆ ∪B)

This proof depends on syntax axioms:   → wi 4   ∧ wa 358   ≠ wne 2517   ⊆ wss 3258  ∅c0 3551  ∪cuni 3892  ∩cint 3927

This proof depends on 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 proof 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-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-ss 3260  df-nul 3552  df-uni 3893  df-int 3928

This theorem is used by:  rintn0  4057