Theorem intssuni2 4923

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 4920	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
2		uniss 4866	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵)
3	1, 2	sylan9ssr 3945	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ≠ wne 2929   ⊆ wss 3898  ∅c0 4282  ∪ cuni 4858  ∩ cint 4897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-v 3439  df-dif 3901  df-ss 3915  df-nul 4283  df-uni 4859  df-int 4898

This theorem is referenced by:  rintn0  5059  fival  9303  mremre  17508  submre  17509  lssintcl  20899  iundifdifd  32543  iundifdif  32544  bj-ismoored2  37173  bj-ismooredr2  37175  ismrcd1  42815