Theorem intssuni2 4901

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 4898	. 2 ⊢ (𝐴 ≠ ∅ → ∩ 𝐴 ⊆ ∪ 𝐴)
2		uniss 4844	. 2 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝐴 ⊆ ∪ 𝐵)
3	1, 2	sylan9ssr 3931	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ⊆ ∪ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   ≠ wne 2942   ⊆ wss 3883  ∅c0 4253  ∪ cuni 4836  ∩ cint 4876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-uni 4837  df-int 4877

This theorem is referenced by:  rintn0  5034  fival  9101  mremre  17230  submre  17231  lssintcl  20141  iundifdifd  30802  iundifdif  30803  bj-ismoored2  35206  bj-ismooredr2  35208  ismrcd1  40436